This video covers a lot of material on quantum mechanics, including its history and some of the experiments that led to its development. It also introduces complex numbers and demonstrates how to solve equations involving them.
00:00:00 Quantum mechanics is a theory that explains the behavior of matter and energy on a very small scale. It is necessary for understanding the behavior of light and electricity, two of the most important aspects of modern life.
00:05:00 This video discusses some difficult experiments in quantum mechanics that don't fit with what was known in classical physics. The final difficult experiment to explain is the bright line spectra for example, when a flame is emitting light with a specific frequency. In classical physics, this light would look like a black body, but the experiment shows that this is not the case. Instead, short wavelengths emit more energy while long wavelengths emit less. These experiments also involve light and matter, and together they explain the Rayleigh jeans law and the photoelectric effect. However, these laws don't always match reality, which is called the ultraviolet catastrophe. This phenomenon is explained by the opposing electric fields produced by a voltage just high enough to stop the electron's motion.
00:10:00 This photograph shows many of the key contributors to quantum mechanics, including Albert Einstein, who developed the theory, and Marie Curie, who discovered nuclear radiation. Classical physics does not explain the spectra seen in these photographs, and quantum mechanics is needed to explain them.
00:15:00 This video introduces the history and work of some of the most brilliant scientists of the 20th century, including Albert Einstein and quantum physicist, John von Neumann. quantum mechanics is a counterintuitive subject, and for beginners, it can be confusing to understand the difference between classical and quantum mechanics. The video explains that in classical physics, everything is known with precision, while in quantum mechanics, what is predicted are probabilities. The boundary between classical and quantum mechanics is not always clear, and for most applications, quantum mechanics is relevant.
00:20:00 In quantum mechanics, the wave function is a mathematical description of the state of a particle. The wave function describes the probability of finding a particular particle at any given point in space and time. It is important to understand quantum mechanics in order to understand how particles behave in the real world. One example of a particle that has quantum mechanics relevant to it is a speck of dust in a light breeze.
00:25:00 In quantum mechanics, the wave function represents the state of a system, but it doesn't always give you information about the observable properties of the system. In addition, operators connect wave functions to observable quantities.
00:30:00 This 1-paragraph summary of quantum mechanics explains that complex numbers are essential to understanding quantum mechanics, as well as the wave function, which is expressed in terms of complex numbers.
00:35:00 Complex numbers are a way of representing real numbers that are not simple. The basic operations of addition, subtraction, multiplication, and division are all represented in complex form.
00:40:00 In this video, the presenter explains how to simplify complex numbers in polar form, which makes them easier to multiply and divide. Additionally, the presenter explains how to calculate the absolute value of a complex number using the complex conjugate and Cartesian coordinates.
00:45:00 In this video, the presenter demonstrates how to simplify complex expressions involving multiplication and division by using the complex conjugate of both the numerator and denominator.
00:50:00 This video explains how to solve equations involving complex numbers. First, the real and imaginary parts of the numerator and denominator are separated, and then the real part of the numerator is divided by the real number that represents the complex number's magnitude. Next, two equations are generated, and solving for x in one of them leads to the equation x equals zero.
00:55:00 In this video, a quantum physics full course is reviewed, which includes a discussion of quantum mechanics. The course begins with an introduction to complex numbers and how to write and solve equations involving them. Two solutions to a cubic equation are found, one involving a real number and the other involving a purely imaginary number. The real number solution is found by solving for x, which equals 1 when cubed, while the imaginary number solution is found when y equals the square root of 3 fourths, which is equivalent to saying y equals plus or minus the square root of three fourths.
This video covers the basics of quantum mechanics, explaining the wave function, probability, and the normalization requirement for a probability distribution. It also discusses the mean squared deviation and skewness of a distribution, and how they relate to the wave function.
01:00:00 This video covers the basics of quantum mechanics, including the wave function and probability. It explains that the wave function corresponds to an observable, and that probability arises from the wave function's uncertainty.
01:05:00 The wave function describes the probability of finding a particle at a particular location. The wave function before a measurement is broad, and the wave function after a measurement is narrow, due to the uncertainty of the measurement.
01:10:00 In quantum mechanics, a wave function collapse is a sudden decrease in the probability of an event occurring. This decrease in probability is due to a measurement being made. The many worlds interpretation of quantum mechanics suggests that when a measurement is made, the universe splits into two different universes.
01:15:00 The statistical term "sigma squared" is used to describe the variability of a continuous distribution. The term "mean squared deviation" is used to describe the variability of a discrete distribution.
01:20:00 This video discusses the variance and skewness of probability distributions. Skewness measures how much a distribution deviates from being symmetrical, while variance measures how much the distribution varies from the mean. The mean squared deviation is a measure of how much a distribution deviates from the mean.
01:25:00 In this video, we learn about skewness and kurtosis, two common measures of statistical shape. We also learn that skewness is related to the integral of x to the fourth row of x dx, and kurtosis is related to the integral of x to the fifth row of x dx. These properties are important for understanding the probability distribution of a wave function. Finally, we discuss the normalization requirement for a probability distribution, which is necessary for an accurate statistical interpretation of the wave function.
01:30:00 Quantum mechanics deals with the behavior of particles in the universe that are smaller than atoms. In order to calculate the behavior of these particles, one must first understand the mathematics of quantum mechanics. If a particle obeys the laws of quantum mechanics, then its behavior can be described by a function that is not normalizable. This means that the function cannot be divided algebraically by a constant, and the area under the graph of the function will always be 1. If a particle solves the Schrodinger equation, then it can be said to have an energy that is a constant multiplied by the particle's mass.
01:35:00 The student explains how to use the schrodinger equation to find partial derivatives with respect to time. He then calculates the partial derivatives of psi with respect to position and time, which are found to be -i h bar over 2m and -i v over h bar, respectively.
01:40:00 This transcript excerpt explains how to integrate the second partial derivative of a function with respect to x, using the complex conjugate of the function. The partial derivatives are then multiplied together, and the result is then integrated. The final result is that the effective total probability of a function at a particular time is equal to the product of the partial derivatives with respect to x.
01:45:00 In this video, the fundamental theorem of calculus is discussed, and it is shown that the integral from minus infinity to infinity of the squared absolute magnitude of psi as a function of both x and time is equal to a constant. It is also shown that normalization constants, such as psi star, are purely real and do not depend on the time evolution of the system.
01:50:00 In quantum mechanics, the wave function psi is related to physical quantities such as position, velocity, and momentum. The mathematical calculation of the expected value of x given a probability density function is an important concept. If we measure the position of the particle to be "here" the new wave function will have to accommodate this new probability density. The observation process, which alters the wave function, means that we can't ever observe an object exactly as it is in reality.
01:55:00 Quantum mechanics includes the mathematical description of the behavior of particles in a system. In order to predict the position of a particle over time, quantum mechanics must account for the effect of a single measurement on the system. This is done by calculating the time derivative of the expected value of position.
This video covers the basics of quantum mechanics, including the wave function, operators, and the uncertainty principle. The course also covers the solution to the Schrodinger equation using separation of variables.
02:00:00 In this video, the presenter explains how integration by parts can be used to simplify an equation involving derivatives. This process is repeated for the second integral, which also vanishes due to the boundary terms. Ultimately, all that is left is the integral of psi star with respect to x.
02:05:00 The video discusses the mathematical expectation of a velocity vector, which is represented by the equation hbar over m times the integral of psi star partial derivative of psi with respect to x. The equation is written in a more suggestive way by saying x hat equals x times something, which is usually written as x equals x times the operator. The equation also has the same expression for the position operator, which is the expected value of position without the hat.
02:10:00 This lecture discusses the uncertainty principle, which is a key result of quantum mechanics. This principle states that the position and wavelength of a wave cannot be measured with absolute certainty.
02:15:00 Quantum mechanics is the branch of physics that studies the behavior of matter and energy as waves. In the early development of quantum mechanics, de Broglie noticed that the spectrum of the hydrogen atom looked similar to a wave. He reasoned that the electrons were travelling around the nucleus of the atom as waves, and that only an integer number of waves fit the spectrum. This insight led to the development of quantum mechanics. The bottom line is that waves and matter waves are related, and that the spread in wavelengths and positions is always greater than about equal to one.
02:20:00 Quantum mechanics is a branch of physics that deals with the properties of matter and waves. The theory of quantum mechanics is based on the principle of uncertainty, which states that the position, momentum, and energy of objects are uncertain. This uncertainty is related to the waves that make up matter and is quantified by the uncertainty principle. In this video, the uncertainty principle is explained, along with the different ways in which it impacts the wave function. It is also noted that the uncertainty principle is fundamental to understanding the validity of quantum mechanics.
02:25:00 In quantum mechanics, the wave function psi is a mathematical representation of the probability of observed outcomes of measurements. The wave function can be thought of as a snapshot of the quantum state of a particle at a certain point in time. The wave function can be decomposed into its constituent parts, which are denoted by operators. The wave function and operators together give us information about the particle's position and momentum. The wave function can be collapsed, which gives us a more definite prediction of an observed outcome.
02:30:00 The quantum mechanics course covers the basics of the theory, including the collapse of the wave function and operators. The course also covers the effects of operators on the wave function, including the position operator and the momentum operator. Finally, the course discusses the expectation values of operators, and how they are similar to the expectation value of the position.
02:35:00 The first thing to understand about solving equations like the Schrodinger equation is that they are partial differential equations. Partial differential equations are more difficult to solve than ordinary differential equations, but they allow for more specific information about how coordinates change with time. In an ordinary differential equation, the position, velocity, and other coordinates all change at the same rate. In a partial differential equation, the rate at which one coordinate changes may be different from the rate at which another coordinate changes.
In order to solve the Schrodinger equation, we need to know about separation of variables. Separation of variables is a solution technique that we will be using repeatedly to solve the Schrodinger equation. First, we will talk about ordinary differential equations and what they are. Next, we will talk about partial differential equations and what they are and what they mean for the Schrodinger equation. Finally, we will discuss how to solve the Schrodinger equation using separation of variables.
02:40:00 In physics, a partial differential equation is a mathematical equation that describes the change in a function over a spatial region. The equation is composed of a set of independent variables (x, y, z) and a set of dependent variables (v). Solutions to these equations are often found by separation of variables, which is a method of solving equations that relies on making assumptions about the behavior of the dependent variables.
02:45:00 In this video, a quantum physics full course, the author discusses the concept of forces and how they are related to curvatures on a string. The author then introduces the wave equation, which is a key equation in quantum physics. The author explains that the acceleration as a result of the force is related to the curvature of the string and how derivatives are used to calculate this. Finally, the author shows how derivatives can be used to divide both sides of an equation by a constant, resulting in a non-trivial solution.
02:50:00 In physics, the separation of variables is a technique used to solve a partial differential equation. In order to use this technique, both the function of interest (e.g. t in the case of a theoretical equation representing a wave) and the function of change (e.g. x in the case of a theoretical equation representing a physical object's position) must be independent variables. In this example, the two independent variables are x and t, and the relationship between x and t is held constant for a certain value of t for a certain value of x. If x is changed without changing t, the left-hand side of the equation is not changing, meaning that g of x is a constant. This equation can be represented in terms of an ordinary differential equation, in which t is only a function of little t.
02:55:00 In a quantum physics course, the separation of variables equation is used to solve partial differential equations, such as the wave equation. This equation is written in terms of the hamiltonian operator, which is related to the total energy of the system. When the equation is solved, the time derivative of the big t variable is found to be minus h bar squared over 2m times the same thing when the equation is solved for x t.
This video discusses the Schrodinger equation and how it governs the behavior of particles in a system. The equation is linear and can be solved using the solution method of separation of variables. If a particle in a system is in a particular stationary state, then the energy of the particle can be found by summing the energies of all the stationary states of the particle. This is an example of an analysis that can be done using superpositions of stationary states. The video also discusses potential energy and how it influences the wave function of a system.
03:00:00 In this video, the partial derivatives of big x with respect to position and time are shown. The first derivative of big t with respect to time is equal to e, and the second derivative of big x with respect to position is equal to the energy.
03:05:00 In quantum mechanics, the Schrodinger equation is a pair of differential equations that govern the evolution of a system's wave function. The time-independent Schrodinger equation describes the evolution of the wave function over time, while the time-dependent Schrodinger equation describes the evolution of the wave function as a function of position. The wave function is a mathematical representation of the system's state. The solution to the time-independent Schrodinger equation is a set of spatial coordinates, while the solution to the time-dependent Schrodinger equation is a set of temporal coordinates. The wave function can be thought of as a "map" of the system's state.
03:10:00 This lecture discusses the properties of stationary states that result from the separation of variables equation. These states are called stationary because they do not change as time evolves. The properties of these states include the magnitude of the squared magnitude of the spatial and time parts of the wave function, which is 1.
03:15:00 The linearity of the Schrodinger equation allows for the simultaneous solution of two equations, thereby allowing for the generation of superpositions of stationary states. These stationary states can exist because the linearity of the Schrodinger equation guarantees that the potential energy of a system remains constant across time.
03:20:00 The video discusses the Schrodinger equation, which is a partial differential equation that describes the behavior of particles in a system. The equation is linear and can be solved using the solution method of separation of variables. If a particle in a system is in a particular stationary state, then the energy of the particle can be found by summing the energies of all the stationary states of the particle. This is an example of an analysis that can be done using superpositions of stationary states.
03:25:00 In quantum mechanics, potential energy is a key factor in determining the likelihood of a particle's location. If the potential energy of a state is high, the particle is less likely to be found there, and the state's potential energy is lower.
03:30:00 In this video, a potential function is described, and the different types of potentials that can be found in quantum mechanics are mentioned. The particle in a box potential is a canonical example of a potential, and the delta function potential is another example.
03:35:00 In this video, quantum physics instructor Michael Nielsen explains the various quantum mechanical states that are allowed by a potential, and how these potentials influence the Schrodinger equation and its solutions. He goes on to discuss what happens when there is no potential, and how this potential function has implications for the form of solutions to the Schrodinger equation.
03:40:00 In quantum mechanics, the wave function describes the probability of an event occurring. If a boundary is crossed, the wave function will change depending on the value of the potential energy at that point. If the potential energy is greater than the energy of the state, the wave function will curve away from the axis and look like a wave. If the potential energy is less than the energy of the state, the wave function will curve towards the axis and look like a wave. The wave function is essentially smooth and will never have any sharp corners.
03:45:00 In this video, a conceptual framework is given for understanding what potential means in quantum mechanics. For a given potential energy, a wave function will have certain behaviors, such as curvature and direction of curve. The potential energy of a system is quantified by the potential function. If the potential energy is larger than the energy of the state, the wave function will curve upwards and away from the axis. If the potential energy is less than the energy of the state, the wave function will curve towards and always curve towards the axis.
03:50:00 The wave function for a particle in a one-dimensional harmonic oscillator is continuous, and obeys the boundary conditions that specify the form of the solution.
03:55:00 The discontinuity in the momentum operator at x = 0 can lead to strange results in quantum mechanics. For example, the momentum operator might blow up, and the wave function might be continuous, meaning that it goes through zero everywhere.
Quantum mechanics is a complex field of physics that can be difficult to understand. This video provides a full course on quantum mechanics, explaining different concepts in detail. These concepts include the time-independent Schrodinger equation, stationary states, the probability density function, and the Commutator.
04:00:00 In this lecture, Professor John Baez discusses the properties of solutions to the time-independent Schrodinger equation, which include orthogonality and completeness. This information is useful in understanding the wave function for a system, as well as calculating the energy associated with it.
04:05:00 In this video, Professor John Baez discusses the concepts of orthogonality and quantum mechanics. He explains that in two dimensions, the dot product of two vectors is zero if the vectors are orthogonal. In three dimensions, the dot product is a sum of the components of each vector multiplied together. In quantum mechanics, complex functions need to be conjugated in order to make sense. In this example, a particle is in a box and its potential energy is described by an infinite square well potential. The stationary state wave functions and energies of the particle can be determined by solving the time-dependent Schrodinger equation.
04:10:00 In this lecture, Professor John Baez discusses how to solve the time-independent Schrodinger equation. He introduces the concept of stationary states and shows how to find the constants in terms of these states' wave functions. Finally, he provides an example of how these wave functions can be plotted.
04:15:00 In this video, the presenter explains how to do integrals in physics using software called sage. The presenter demonstrates how to evaluate c of n for x going from one to ten.
04:20:00 Quantum physics is a complex and fascinating field that can be explained in terms of simple mathematical expressions. This video discusses how to approximate these expressions using a software such as sage.
04:25:00 In this video, the presenter describes the probability density function (PDF) for a given wave function. He shows how to calculate the PDF for a given wave function, and discusses how it changes over time.
04:30:00 The video discusses the time variability of a quantum mechanical system, which can be seen in the curves representing the wave function and its probability density. The wave function gradually expands and narrows over time, as well as oscillating around a larger value or zero. It also shows how the amplitude and phases of a particular stationary state can be seen.
04:35:00 In this video, a quantum physics full course, quantum mechanics is explained in detail. Different types of stationary states are explored, along with the effects of time variability and excited states. Solutions to the Schrodinger equation are explained, and the importance of ladder operators is highlighted.
04:40:00 The video discusses the concept of a spring and its potential energy. The potential energy is a function of x, and is calculated as the half of the spring's potential energy, multiplied by the displacement x squared. The traditional way to write this potential is in terms of the angular frequency of the oscillations that result when a mass is on a spring, using the square root of the spring constant divided by the mass of the particle. However, one can also solve the Schrodinger equation for harmonic oscillators under certain circumstances, and these solutions will look like a hump-shaped wave function. In this case, one can use cleverness to reduce the number of calculations needed to get a harmonic oscillator solution, by factoring out the hamiltonian operator.
04:45:00 In this video, a quantum operator is defined, and its behavior is discussed. It is shown that, unlike numbers, operators do not always behave in the same way, and that in order to factor an operator, one must first simplify it using parentheses and a hat. It is demonstrated that, in this case, the hamiltonian is the result of multiplying two operators, and that its interpretation as a momentum vector is suggestive.
04:50:00 The Commutator in quantum mechanics is a mathematical operation that describes the relationship between two operators. It is defined for two operators, a and b, and is represented by x hat p hat minus p hat x hat . This equation allows for the operators to act on each other.
04:55:00 In this video, the operator algebra that was used to factor the hamiltonian is explained, and the cleverness comes in when considering ladder operators and energy. The time-independent Schrodinger equation is solved, and the hamiltonian can be expressed in terms of these ladder operators, a plus a minus operators, and a half. The wave function is expressed in terms of these operators and the hamiltonian, and it is shown that if a plus is allowed to act on the wave function before the hamiltonian is applied, then the expression can be rewritten as a plus acting on the hamiltonian.
This video explains how to solve the wave equation for a quantum system. It discusses the various constants involved in the equation, and how to use the chain rule to simplify the calculation. The video also explains how to use the Fourier transform to analyze the wave function.
05:00:00 In this video, quantum physics professor John Baez discusses how the ladder operator, a plus sign, can be used to generate solutions to the Schrodinger equation. This process can be repeated indefinitely, generating an infinite number of solutions with increasingly high energy.
05:05:00 In quantum mechanics, the wave function is a mathematical model that describes the state of a system. If energy is lowered to a low enough level, the wave function will no longer be able to provide a meaningful solution, and the system will be in a state of lowest energy. This is a problem because it means that the system cannot be in multiple states at the same time. In order to solve the system, one must identify a wave function that is capable of providing a solution with low energy. This is done by considering the lowering operator and its relationship to zero. Once the wave function is identified, the equation governing its behavior can be solved. This process results in the formation of multiple states of the system, which can be explored by applying the raising operator.
05:10:00 The video discusses quantum mechanics, and explains the normalization constant, psi sub 0, and psi sub n. It goes on to explain how to calculate the energies of the solutions to a wave equation, and ends with an example calculation of psi one.
05:15:00 This video covers the basics of quantum mechanics, including the following constants: m omega over pi, h bar to the one fourth power, square root of two m omega over h bar, x e to the minus m omega, x squared, and psi one. The formulas for these terms can be very complicated, but by manipulating derivatives and applying the chain rule, the formulas can be simplified to give a wave function for the quantum harmonic oscillator. The equation for the wave function can be solved using power series, which is a common solution technique for ordinary differential equations.
05:20:00 This video covers the chain rule and how to use it to solve derivatives of a function with respect to another function. The chain rule can be applied to a derivative term with two first derivatives, a second derivative, and a constant. The final equation is a second derivative of a function with respect to a constant.
05:25:00 The approximate equation for large values of c in the Schrodinger equation is c squared psi, which is approximately equal to c. This approximation is useful in solving the Schrodinger equation, which is one way that the free particle theory applies.
05:30:00 In this video, the mathematician and physicist, Dr. John Baez, discusses the solution to the time-independent Schrodinger equation, which is a wave function describing the state of a particle at any given time. The solutions are traveling waves, and can be thought of as a traveling wave in the complex plane.
05:35:00 In this video, a quantum physicist explains how waves can be combined to create something that is real, in the form of a wave packet. This packet can be used to solve the Schrodinger equation for the free particle, and can be localized in space or time. However, the wave packet construction is not entirely smooth, and requires some mathematical trickery.
05:40:00 In the quantum physics course, students explore how an integral can collapse when multiplied by a particular stationary state wave function. This analogy is used to explain the collapse of the sum over n in the case of the particle in a box and the free particle.
05:45:00 In this video, the presenter explains how the delta function can be used to calculate the psi of a given x value. This information is important for understanding Fourier analysis.
05:50:00 In this video, the presenter explains how to use the Fourier transform to analyze wave functions. They talk about the wave velocity, and how to find it using the argument. They also discuss how to calculate the fee of k that goes with a particular function.
05:55:00 Quantum mechanics is a branch of physics that studies the behavior of matter and energy on the atomic and subatomic level. In quantum mechanics, particles such as atoms and photons have wave-like properties, which means they can be described by a wave equation. This wave equation describes the behavior of the particles, and it can be solved to determine the velocity of a particular feature on the wave. However, this velocity is not the same as the classical velocity of a wave packet. This is because wave packets are the only real states that we can observe in the physical universe, and quantum mechanics deals with the motion of features on waves, not the motion of waves themselves.
This video provides an introduction to quantum mechanics, covering topics such as wave packets, the Dirac delta function, and the time independent Schr¨dinger equation. It also discusses the solutions to the free particle potential equation and the equation x equals zero for a wave function.
06:00:00 In quantum mechanics, wave packets represent the collective behavior of particles. The velocity of a wave packet is determined by its average k and its flux (or momentum).
06:05:00 The video explains the wave packet velocity equation, which states that the speed of a wave packet is approximately equal to the average energy divided by mass in the square root. This is not the classical equation, which states that the speed of a wave packet is determined by the energy of a single wave. The wave packet velocity equation describes the behavior of a wave packet as it propagates through space.
06:10:00 In this video, a student explains how the Dirac delta function works mathematically and gives an example of its limit as sigma gets smaller. This function gets narrower and taller as sigma gets smaller, and its dependence on x squared gets faster.
06:15:00 The delta function is a useful tool for calculating the expected value of a function at a specific point, and can be used in conjunction with other functions.
06:20:00 In this video, the author explains the concept of a step discontinuity in a wave function, and how it is related to the second derivative of a delta function. He then goes on to discuss how to solve the time independent Schrodinger equation for a potential given by a delta function.
06:25:00 In this video, the solutions to the free particle potential equation are described, including how they change depending on the energy of the solution. The square root of a negative quantity, k, is introduced as the constant for the potential. Finally, the solutions are explained in terms of psi of x, a function of x and k.
06:30:00 In this video, the author explains the solution to the equation x equals zero for a wave function that is normalizable. To solve for this solution, the author uses the mathematical equation for delta functions and integrates over the boundary to find the value of psi at the boundary.
06:35:00 In this video, a student explains the integral of a second derivative with respect to x of a Schrodinger equation on the boundary of two regions, identifying regions and calculating the values of the first derivatives of psi. On the right hand side, this results in the expression b k e to the plus or minus kx, which becomes 1 when evaluated at zero.
06:40:00 In this video, quantum physics instructor John Baez explains the concepts of quantum mechanics and how these concepts relate to the behavior of particles in a box. He goes over the concept of quantization, which occurs when a wave function has finite energy and can only take on certain discrete values. He also discusses the concept of a bound state solution, which is the most general possible form of the wave function and represents the most stable configuration of the particles. Finally, he discusses the scattering state solutions, which are solutions that are away from the delta function potential and can be matched together using boundary conditions.
06:45:00 Quantum physics can be complex, but this course walks you through the basics of quantum mechanics. In particular, the course covers the following:
-The basics of quantum mechanics, including the energy of a particle and the solutions it can take.
-The behavior of a particle at a boundary.
-Two general solutions for a particle in a region away from a delta function.
-The boundary condition for a particle at a boundary.
-How to match solutions for a particle at a boundary.
06:50:00 Quantum physics can be explained in terms of Schr¨dinger's equation and the boundary condition that governs the behavior of the first derivative of psi. The first derivative of psi is given by ps1 = ae+bE, where a and b are constants, and psi is at the boundary. If x equals zero, then psi1 and psi2 are both positive, and f and g are negative. If x equals infinity, then psi1 and psi2 are both negative, and f and g are both positive. When x equals zero, the two equations become one, and when x equals infinity, psi1 and psi2 become zero. When x equals some value in between, the two equations become two, and the two unknowns a and b must be solved for.
06:55:00 In this video, a quantum physics full course is discussed. The lecture covers the positive values of k and the wave propagating to the right. Solutions to the Schrodinger equation are found for three regions: between minus a and a, greater than minus a, and everywhere above the potential. The free particle solution is used to explain traveling waves. When adding the time dependence, the solutions are found to be traveling waves.
This video provides a detailed introduction to the mathematical formalism of quantum mechanics, including the concepts of eigenvalues, eigenstates, and hermitian operators. It also discusses the properties of orthogonality and completeness, and how they can be used to represent any arbitrary state.
07:00:00 Quantum physics provides a detailed understanding of the behavior of matter and energy on a very small scale. In this video, three solutions to the Schrodinger equation are described, with the help of boundary conditions and sound physical reasons. One solution is called psi and is described as the c-th power of the sine of the l-th coordinate multiplied by d-th power of the cosine of the l-th coordinate. The second solution is called psi-two and is described as the c-th power of the sine of the l-th coordinate multiplied by the cosine of the l+1-th coordinate. The third solution is called psi-three and is described as the c-th power of the sine of the l-th coordinate multiplied by the cosine of the l+2-th coordinate. Meshing these solutions together with desired boundary conditions results in a final solution that is equal to psi-two plus psi-three.
07:05:00 In this video, quantum physicist James Gill discusses how linear algebra can be used to simplify quantum mechanics problems. He explains that, in quantum mechanics, waves are scattered off of a delta function potential, and that linear algebra can be used to simplify the algebra. He then goes on to say that, in quantum mechanics, each of these expressions in terms of vectors has an analog. Finally, he discusses how quantum mechanics can be used to justify why linear algebra is useful.
07:10:00 In this video, a whirlwind tour of linear algebra is given, including linear transformations with hats. Linear algebra is used to represent the state of a quantum mechanical system, and the state is everything that we can possibly know about the physical system. The state can be represented as a superposition of wave functions, and the wave function can be represented as an integral.
07:15:00 In quantum mechanics, states are represented by vectors in Hilbert space. These vectors have an inner product that is equal to 1, and normalization and orthogonality are also maintained. Linear algebra concepts, such as the inner product, are still applicable.
07:20:00 In quantum mechanics, an operator is a mathematical function that acts on the state of a quantum system. The expectation value of an operator is a real number that corresponds to the probability of observing a certain state of the system.
07:25:00 In this video, quantum physicist James Hartley explains how complex conjugated variables, such as q hat psi, can be represented in mathematical terms using the complex conjugate of an expectation value. He also discusses the importance of hermitian operators in quantum mechanics, and demonstrates how the momentum operator behaves when acting on a complex conjugate of a state.
07:30:00 In this video, a Quantum Physics Full Course is discussed, including the concept of minus i h bar partial g partial x, which is a hermitian operator. This operator is required by the notion that the momentum operator be hermitian, and is also required to represent an observable with no uncertainty. The state of determinate energy solutions to the time independent Schrodinger equation are also mentioned.
07:35:00 This 1-paragraph summary explains the mathematical formalism of quantum mechanics, and how hermitian operators can be used to solve eigenvalue problems. This can be applied to various observable quantities in quantum mechanics, such as the energy of a quantum harmonic oscillator, and the positions and momenta of particles.
07:40:00 The eigenvalues of hermitian operators are real, and can represent observables. The eigenvalues can be found by applying the operator on the left to the wavefunction, and the eigenvalue on the right to the state.
07:45:00 In this video, the instructor discusses the eigenvalues and eigenstates of a hermitian operator. If the eigenvalues are not equal, the inner product between the states can be zero, which causes a problem. The instructor discusses a method for ensuring that the eigenstates are orthogonal to each other, known as gram-schmidt orthogonalization.
07:50:00 In quantum mechanics, orthogonality is a mathematical property of states that allows for the unrestricted use of mathematical operators on these states. Completeness is a property of a quantum mechanical state space that allows for the representation of any arbitrary state as a superposition of a set of stationary states.
07:55:00 The video discusses the mathematical definition of orthogonality, and shows how it can be used to represent any arbitrary initial conditions. It also explains how to calculate the delta of x, or the difference between x and c.
In this video, the presenter discusses the various concepts behind quantum mechanics, including the wave-particle duality and the Schr¨dinger equation. He then goes on to discuss eigenstates and eigenvalues, showing how these concepts are related to the operator q and the momentum operator. He concludes the video by discussing how quantum mechanics can be used to calculate the expected values of various physical quantities.
08:00:00 Quantum mechanics revolves around the mathematics of hermitian operators and eigenvalues. The spectrum of an observable can be determined by Fourier's trick, and the completeness of a basis can be expressed as an integral. In the context of uncertainty, statistical properties of a quantum mechanical system can be determined by applying the formal mathematical structure to an observation.
08:05:00 Quantum mechanics is a theory that describes the behavior of matter and energy on a very small scale. The theory states that when particles are observed, their state (location and momentum) is determined by a wave function that contains information on all possible outcomes. The wave function is represented by a series of basis vectors in Hilbert space, and the probability of a particular outcome is given by the magnitude of the wave function of q squared multiplied by the probability of that particular outcome.
08:10:00 In this video, quantum physics instructor John Baez discusses the orthogonality relationship and how it collapses two sums together to get one final sum. He also explains how normalization conditions imply that the sum of the squares of the coefficients in a state's representation is one. Finally, he demonstrates how this same logic can be used to compute expectation values for general operators.
08:15:00 In this video, the presenter discusses the various concepts behind quantum mechanics, including the wave-particle duality and the Schr¨dinger equation. He then goes on to discuss eigenstates and eigenvalues, showing how these concepts are related to the operator q and the momentum operator. He concludes the video by discussing how quantum mechanics can be used to calculate the expected values of various physical quantities.
08:20:00 Quantum mechanics is a field of physics that deals with the behavior of matter and energy on the atomic and subatomic level. In this video, the instructor discusses the probability of observing various energies and states of a particle. The instructor also discusses the uncertainty principle, which states that the magnitude of an observable's uncertainty is greater than the uncertainty in its respective Observer's knowledge of that observable. Using linear algebra, quantum mechanics allows for the construction of probabilistic interpretations of observables beyond just position and momentum.
08:25:00 The uncertainty principle states that two observables, such as position and momentum, cannot be simultaneously known with absolute certainty. Quantizing these observables into a finite number of quanta (i.e. bits) allows for the calculation of their joint uncertainty. This uncertainty can be expressed as a product of the squared deviations of the two observables from their respective expected values.
08:30:00 The "Schwarz inequality" states that the inner product of two vectors is always greater than or equal to the absolute magnitude of their respective vectors. This inequality is simplified in terms of the magnitude of their inner product. The second simplification is that the imaginary part of a complex number is always greater than or equal to the real part.
08:35:00 In this quantum physics video, the author explains how to calculate the expectations of two operators, psi and qhat, using complex numbers. This calculation is necessary to get the main result, which is the expectation value of qhat r hat minus mu q.
08:40:00 The generalized uncertainty principle states that two observables have a maximum uncertainty if their commutator has a non-zero expectation value. This principle is based on the Schwartz inequality, which states that the inner product of a vector with itself is greater than the squared modulus of the inner product of the vector with another.
08:45:00 In this video, quantum physics instructor John Schwartz discusses how to turn inequalities into equalities using vectors and operators. He explains that if you throw out the real part of a complex number, the magnitude of the complex number will only change when you throw out the real part. This means that the two inequalities become equalities when the real part of the complex number is zero. This has implications for the concept of position and momentum uncertainty.
08:50:00 The video discusses the Generalized Uncertainty Principle, which states that the uncertainty principle applies to any two quantum mechanical operators. This principle is a powerful tool that can help physicists understand the behavior of quantum mechanics systems.
08:55:00 The video explains the difference between position, momentum, and energy uncertainty principles in quantum mechanics. It also introduces the time derivative of the expectation value of a quantum operator, qhat. This expression can be simplified using the time dependent Schrodinger equation.
This video provides a full course on quantum mechanics, discussing topics such as operators, angular momentum, and ladder operators. The author also explains how to calculate the commutator of two operators and how to find eigenstates.
09:00:00 In this video, a simplified derivation of the quantum mechanics equation of motion is presented. The equation of motion is reduced to a hermitian operator equation, and the constants are simplified by multiplying by a constant on the right and moving the variable inside the equation. The operator equation is then simplified by factoring out the psi term on the left and the psi term on the right. The expectation of the commutator is then obtained, resulting in a minus 1 over i h bar psi.
09:05:00 The uncertainty principle states that the energy uncertainty of a system is related to how quickly it changes.
09:10:00 This video discusses the conventional energy time uncertainty relation, which states that the uncertainty in energy (delta e) is greater than equal to h bar over 2. This relation is valid for any observable q, which means that the energy uncertainty will be small if the system is evolving rapidly and large if the system is relatively stable.
09:15:00 In this video, a course on quantum physics, the basics of one-dimensional quantum mechanics are covered. In three dimensions, the same mathematical problems exist, but with more complex operators. Finally, quantum mechanics is applied to the real world, with examples of how it works in three dimensions.
09:20:00 This video introduces the principles of quantum mechanics, including spectroscopy and the energy levels of atoms. It explains how the principles of quantum mechanics can be used to calculate the energy of photons and atoms.
09:25:00 In this video, the author provides a full course on quantum physics, including explanations of energy levels, transitions, and spectroscopic lines. The author also discusses the observable spectrum of hydrogen, and explains how quantum mechanics can't always predict the behavior of more complex atoms.
09:30:00 In this video, the author discusses the concepts of quantum mechanics, including the level of complexity involved, and introduces the concepts of angular momentum and momentum operator. He goes on to show how angular momentum can be represented in terms of operators, and how this can be used to solve problems in quantum mechanics.
09:35:00 In this video, the author explains how angular momentum is quantized in three dimensions. He describes the commutators of angular momentum operators, and how multiplying them together gives the angular momentum operator.
09:40:00 The video demonstrates how to calculate the commutator of various operators, including the commutator of lx and ly. The commutator of lx and ly does not commute, meaning that if you want to determine simultaneously lx and ly, you have to consider the uncertainty relation between lx and ly.
09:45:00 The commutator of lx and lz is minus i h-bar l-y, which simplifies the expression.
09:50:00 In this video, the author explains the algebraic properties of angular momentum operators, and how ladder operators can be used to find eigenstates.
09:55:00 In this video, the author discusses quantum mechanics, including the concepts of operators, ladder operators, and eigenvalues. He goes on to explain how to calculate the commutator of two operators, and how eigenfunctions can be obtained for an operator acting on a wave function in general polar coordinates. However, in this lecture, the author runs into some difficulties with notation, and is unable to express the angular momentum operators in spherical coordinates.
The video explains how to find the angular momentum of a particle in spherical coordinates, and how to solve the eigenvalue problem to find the wave function for the angular momentum. The video also explains how spin is an intrinsic property of certain particles and how it can be described using the same language as angular momentum.
10:00:00 In this video, the author reviews the angular momentum operators in spherical coordinates. These operators are written as r cross p, where r is the position operator, r cross r is the momentum operator, and r cross theta is the torque operator. The gradients of these operators are expressed in spherical coordinates as r cross d theta. Finally, the angular momentum operator is written as l plus or minus i h bar.
10:05:00 The video describes how to find the x, y, and z components of the angular momentum in spherical coordinates using the angular momentum operator in cartesian coordinates. The eigenvalue problem is solved to find the wave function for the angular momentum.
10:10:00 This 1-paragraph summary covers the basics of angular momentum and its associated mathematics. angular momentum is a vector quantity that describes the rotation of an object, and is related to the mass and moment of inertia of the object. Half integer values of angular momentum have physical significance in the context of spin angular momentum.
10:15:00 In this video, the author explains quantum mechanics, focusing on the concept of angular momentum. He explains that angular momentum is a conserved quantity in classical and quantum mechanics, and that half integer values of angular momentum are fully valid physical solutions. He goes on to discuss spin, explaining that it is an intrinsic property of certain particles and that it can be described using the same language as angular momentum.
10:20:00 Quantum mechanics is a branch of physics that deals with the behavior of single particles, as well as the behavior of multiple particles. In order to understand quantum mechanics, one must first be familiar with wave functions, eigenfunctions, and the hamiltonian. In addition, one must also be aware of the momentum operators and the gradient operator.
10:25:00 In quantum mechanics, wave functions and the momentum of particles are representations of probability densities. The potential energy is a function of the positions of the particles and must be added to the hamiltonian. The time-independent Schrödinger equation is a partial differential equation in multiple variables, and the solutions to the equation are still the same as before. If particles are indistinguishable, then the wave function looks like a single particle. This is called the principle of indistinguishability of identical particles.
10:30:00 Quantum mechanics tells us that particles are in principle and distinguishable, and that we can only know about their wave function. This wave function is all we can know about the particles, and it specifies their position and other properties. If we exchange the positions of two particles, the wave function for one particle will be equal to the wave function for the other particle. This implies some constraints on the allowable forms of the wave function, which we need in order to encode the indistinguishability of particles into our formulation of quantum mechanics.
10:35:00 In quantum mechanics, the wave function of a system must be invariant under exchange of its constituent particles. This symmetry requirement is known as the indistinguishability of particles. If two particles are exchanged, their wave functions must be identical, thus preserving the system's invariance. This law of physics is essential for the proper functioning of quantum mechanics.
10:40:00 The exchange operator and the hamiltonian operator commute, and this symmetry allows for solutions to be written in wave form. The indistinguishability of particles is an axiom in quantum mechanics. The exchange can't affect the energy of a state, and the polyexclusion principle holds for both fermions and atoms.
10:45:00 This quantum physics video explains how two particles in the same quantum mechanical state cannot occupy the same state. This is due to the anti-symmetric combination of the particles' wave functions.
10:50:00 In quantum mechanics, degeneracy refers to the fact that certain energy levels in a system are allowed to be occupied by the same particle. This can have consequences in the physical world, where two particles that are not distinguishable can share the same energy level. In the case of bosons, this leads to the pauli exclusion principle, which states that no two bosons can occupy the same quantum mechanical state.
10:55:00 This video discusses how quantum mechanics works with particles that are never found outside of a certain rectangular region.
In this video, the instructor explains how quantum mechanics can be used to solve problems involving particles in a box. The starting point is a single particle in a box, and the solution procedure is very similar to the one-dimensional particle in a box. The Schrödinger equation is time-independent and is given by h psi equals e psi. If we make our usual separation of variables assumption, psi is given by a function of x, y, and z. The wave function for a single particle in a three-dimensional box has sine and cosine terms, but the boundary conditions fix our quantization and give us quantum numbers n x, n y, and n z. The allowed energies of the system are given by h bar squared pi squared over 2m.
11:00:00 In this video, the instructor explains how quantum mechanics can be used to solve problems involving particles in a box. The starting point is a single particle in a box, and the solution procedure is very similar to the one-dimensional particle in a box. The Schrödinger equation is time-independent and is given by h psi equals e psi. If we make our usual separation of variables assumption, psi is given by a function of x, y, and z. The wave function for a single particle in a three-dimensional box has sine and cosine terms, but the boundary conditions fix our quantization and give us quantum numbers n x, n y, and n z. The allowed energies of the system are given by h bar squared pi squared over 2m.
11:05:00 Quantum mechanics is the study of the behavior of matter and energy on the atomic and subatomic level. In this course, the instructor demonstrates how to think about quantum mechanics in terms of three-dimensional vectors, which is a useful tool when trying to fill many quantum states.
11:10:00 In this lecture, the professor discusses how quantum mechanics affects the behavior of electrons in solids. He explains that, in order to calculate the potential energy of an electron in a material, one must consider the electron as a wave packet, and make simplifications such as assuming that the potential only depends on the magnitude of the distance between the electron and the atom. He then goes on to describe a dirac comb, which is a potential formed from delta functions.
11:15:00 Blocks theorem states that the wave function for a periodic potential is just equal to the potential at the current location. This is useful for materials where the edge effects aren't very significant.
11:20:00 The video discusses the Schrodinger equation and its boundary conditions. The solutions of the equation are found by matching boundary conditions at two different points, x=0 and x=a. The solutions at these two points are multiplied by a capital K to find the energy of the system.
11:25:00 This video teaches how to solve the continuity and boundary conditions for a wave function, using the delta function potential. After simplifying the equation and eliminating the capital letters, the final equation reduces to a cosine e-to-the-i capital k-a.
11:30:00 The video discusses the equation for the allowed energies of a particle in a quantum state, which is determined by the energy of the state, the strength of the delta function, and the mass of the particle. The allowed energies are packed in a negative and positive range, with no single, isolated value.
11:35:00 In physics, an electron in a material can occupy a number of energy levels, with the lowest energy level being the most free and the highest energy level being the most bound. Electrons in a conductor are free to move between these levels, while electrons in an insulator are not. Temperature affects the number of available energy levels, and a conductor becomes an insulator when all of its energy levels are filled. A semiconductor has few energy levels between the free and bound electron states, and adding or subtracting electrons between these levels can make the material act as a conductor or insulator.
11:40:00 The video discusses the concept of energy bands in materials, and how they are determined by the relative population of energy states. It also describes how to calculate the bands using the wave equation.