In this way, I hope that students gain a physical understanding of the nature of the quantum results. For example, in the study of angular momentum you will find that the absolute squares of the spherical harmonics can be given a relatively simple physical interpretation.
Introduction
Electromagnetic Waves
- Radiation Pulses
- Wave Diffraction
The building block solution of the wave equation is the infinite, monochromatic plane wave solution that has an electric field vector given by Radio waves are also quasi-monochromatic, the central frequency being the frequency of the station (about 1000 kHz for AM and 100 MHz for FM) broadcasting the signal.
BlackBody Spectrum: Origin of the Quantum Theory
In thermal equilibrium at temperature T, the electrons in the cavity walls have a maximum energy of the order of several kBT, so they are only able to excite. Quantitatively, it can be shown that the maximum in the blackbody spectrum as a function of frequency occurs at
![Fig. 1.3 Intensity per unit wavelength (in arbitrary units) as a function of wavelength [in units of (hc=k B T)] for a blackbody](https://thumb-ap.123doks.com/thumbv2/1libvncom/9118262.0/24.659.162.499.88.347/intensity-wavelength-arbitrary-units-function-wavelength-units-blackbody.webp)
Photoelectric Effect
When the intensity of the ultraviolet radiation increases, the number of ejected electrons increases, but the maximum kinetic energy of the ejected electrons does not change. Increasing the intensity of the radiation does not change this maximum kinetic energy, as it is the result of single photon events, it only affects the intensity of the electrons emitted.
Bohr Theory
The quantity is the radius of the orbit, versus the speed of the electron, and is the magnitude of the angular momentum. Bohr's theory also predicts the exact characteristic values for the radius and velocity of the electron in different orbits.
De Broglie Waves
This means that Bohr's theory agreed very well with the experimental values for the frequencies of the emitted lines. On the other hand, if you calculate the de Broglie wavelength of an electron in the ground state of hydrogen, you will find a value close to the Bohr radius.
The Schrödinger Equation and Probability Waves
The interpretation given is that j .r;t/j2 is the probability density (probability per unit volume) of finding the particle at position time t. This will not tell you where the particle will be the next time you do the experiment - only the probability.
Measurement and Superposition States
- What Is Truly Strange About Quantum Mechanics
- The EPR Paradox and Bell’s Theorem
- The EPR Paradox
- Bell’s Theorem
According to quantum mechanics, the spin of each of the particles (by "spin" I now mean either up or down) is not fixed before. In other words, the spin of each of the particles can be either up or down.
Summary
In other words, if Bell's inequalities are violated, then physical observers may not have physical reality of the type described by EPR, regardless of whether or not quantum mechanics is a valid theory. At this level, it's probably best not to worry too much about it and instead focus on mastering the basics of quantum theory.
Appendix: Blackbody Spectrum
- Box Normalization with Field Nodes on the Walls
- Periodic Boundary Conditions
- Rayleigh-Jeans Law
- Planck’s Solution
- Approach to Equilibrium
The frequency that gives rise to the maximum in the energy density is obtained by setting du=d!D0, which yields. One is therefore faced with the modeling of this interaction and then a model for the energy distribution of the charges in the cavity.
Problems
Turning to the modes of the radiation field in the cavity, there is no apparent way that they can exchange energy to reach equilibrium. If the experiment is repeated many times, what will the graph of the number of counts at the detector and detector position look like?
Extra Reading
Mathematical Preliminaries
- Complex Function of a Real Variable
- Functions and Taylor Series
- Functions of One Variable
- Scalar Functions of Three Variables
- Vector Functions of Three Variables
- Vector Calculus
- Probability Distributions
- Fourier Transforms
- Dirac Delta Function
- Problems
In other words, the existence of Fourier transforms leads us to the representation of the Dirac delta function given by Eq. To do this use the definition of the delta function as the limit of a Gaussian,.
Free-Particle Schrödinger Equation: Wave Packets
- Electromagnetic Wave Equation: Pulses
- Schrödinger’s Equation
- Wave Packets
- Free-Particle Propagator
- Summary
- Problems
It is easy to show that equation 3.7) is also a solution of the wave equation, provided!Dkc> 0:Furthermore, since!Dkc,. The variance of a wave packet is its initial variance plus the contribution attributable to the variance of the velocity components in the packet. D 1 for any > 0 is associated with the sharp edges of the initial coordinate spatial wave packet.
Schrödinger’s Equation with Potential Energy
Introduction to Operators
- Hamiltonian Operator
- Time-Independent Schrödinger Equation
- Summary
- Appendix: Schrödinger Equation in Three Dimensions
- Problems
The right-hand side of the free-particle Schrödinger equation in coordinate space is r2, while it ispO2=2min momentum room. I found that the square of the momentum operator is proportional to the Laplacian operator in coordinate space. We have seen that it is possible to obtain a solution of the time-dependent Schrödinger equation if we are able to solve the time-independent Schrödinger equation.
Postulates and Basic Elements of Quantum Mechanics: Properties of Operators
Hermitian Operators: Eigenvalues and Eigenfunctions
- Eigenvalues Real
- Orthogonality
- Nondegenerate Eigenvalues
- Degenerate Eigenvalues
- Completeness
- Continuous Eigenvalues
- Relationship Between Operators
- Commutator of Operators
- Commutator Algebra Relationships
- Uncertainty Principle
- Examples of Operators
The point is that it is always possible to choose N linear combinations of the eigenfunctions. In Chapter 4 I already derived an expression for the square of the impulse operator in coordinate space. We will see later that the fact that the momentum operator commutes with the Hamiltonian is related to the translational symmetry of the Hamiltonian.
Back to the Schrödinger Equation
- How to Solve the Time-Dependent Schrödinger Equation
- Quantum-Mechanical Probability Current Density
- Operator Dynamics
- Sum of Two Independent Quantum Systems
Another way of saying this is that the dynamical variable associated with any Hermitian operator commuting with the Hamiltonian is a constant of the motion. I can guess a solution where the eigenfunctions of the composite system are simply the product of the eigenfunctions of the individual systems, while the eigenenergies are the sum of the individual energies. For two independent systems, the eigenfunctions are products of the individual system eigenfunctions, and the eigenenergies are the sum of the individual system eigenenergies.
Measurements in Quantum Mechanics: “Collapse”
Summary
A detailed catalog has been constructed giving various properties of Hermitian operators and their eigenfunctions and eigenvalues. The central problem of quantum mechanics is reduced to finding the eigenfunctions and eigenvalues of the Hamiltonian operator, from which we can derive all the properties of quantum systems.
Appendix: From Discrete to Continuous Eigenvalues
Problems
If pOx does not commute with the Hamiltonian, how do you know that eipx=„ is not an eigenfunction of the Hamiltonian. How do you know that eigenfunctions of the momentum operator must be eigenfunctions of the free particle Hamiltonian. Why is it that eigenfunctions of the free particle Hamiltonian are not necessarily eigenfunctions of the momentum operator, even though the two operators commute.
Problems in One-Dimension: General Considerations, Infinite Well Potential,
General Considerations
Classically, in case (a), a particle with energy E1 incident from the left would be repelled from the potential. For the potential (Fig. 6.2) (a), the classical particle bounces off the potential, as does the quantum wave packet incident from the left. Quantum mechanically, a wave packet incident from the left and having energy E>W can be reflected.
Infinite Well Potential
- Well Located Between a=2 and a=2
- Well Located Between 0 and a
- Position and Momentum Distributions
- Quantum Dynamics
For large n, the distribution reflects the distribution of the classical distribution given in Eq. 6.4), as it consists of two peaks centered at In other words, I can ask questions like "How many eigenfunctions are needed in the expansion of the initial state wave function?",. The particle creates order. 6.41) wall collisions before scattering of the packet are significant (dB D h=p0 is the average de Broglie wavelength of the initial packet).
Piecewise Constant Potentials
- Potential Step
- Square Well Potential
- Potential Barrier
For physical reasons, the continuity of the wave function is consistent with the idea that the probability density must be a single-valued function. I can estimate the energy EC of the bound state in the limit of the weakly binding cavity, ˇ1. As in the case of the potential well, I consider only the eigenfunction corresponding to the wave incident from the left, namely
Delta Function Potential Well and Barrier
- Square Well with E < 0
- Barrier with E > 0
It follows from the nature of solutions of potential barrier or well problems that the wave function is continuous at the location of the delta function potential. The derivative of the wave function is subject to a point jump discontinuity at the location of the delta function potential. If you consider the transfer problem for a negative delta function potential, the transfer coefficient is unchanged since T depends only on V02.
Summary
Appendix: Periodic Potentials
- Bloch States
Many equations are left random, but all calculations are performed for delta function potentials. The spread of the first band is shown in Figure 6.18, and the real part of the transmission amplitude for this band is shown in Figure 6.19. The end result is that they are in low energy bands where D z0=m 1; transmission resonances resolved and correspond to quasi-bound states.
Problems
- Advanced Problems
For energies E < V0 (the classically forbidden region), prove that the possible solutions of the Schrödinger equation are ex,ex, cosh.x/, sinh.x/, provided D p. The state probability distribution Thek for the eigenfunctions of the infinite square well of widthLis given by [see Eq. Calculate the probability current density of the entire wave function and show that it vanishes in all space.
Simple Harmonic Oscillator: One Dimension
- Classical Problem
- Quantum Problem
- Eigenfunctions and Eigenenergies
- Time-Dependent Problems
- Summary
- Problems
However, you can make some progress towards a solution by building in the asymptotic form of the wave functions. From the form of the Hamiltonian it is clear that the energy is divided equally between kinetic energy (2=2) and potential energy (2=2). Then use the continuity of the wave function and its derivative atD0 to obtain an expression for the reflection coefficient.
Problems in Two and Three-Dimensions
General Considerations
- Separable Hamiltonians in x; y; z
- Free Particle
- Two- and Three-Dimensional Infinite Wells
- SHO in Two and Three Dimensions
- General Hamiltonians in Two and Three Dimensions
- Summary
- Problems
Ifax D ay D az the degeneracy of the modes increases roughly linearly with D q. 8.1.3 SHO in two and three dimensions. It is a trivial matter to solve the SHO problem in two and three dimensions using rectangular coordinates. I have taken a short excursion to look at some simple problems in two and three dimensions.
Central Forces and Angular Momentum
Classical Problem
For now I'm assuming L D Luz so the motion is in the xy plane. The kinetic energy can be rewritten as TD 1. 9.9) was derived assuming that LDLuz, remains valid for arbitrary directions Lif Eq. where.r; ; /are now spherical coordinates. 9.9) is radial, and the second term is angular, or the first term in Eq. 9.9) is radial, and the second term is the contribution of angular or rotational motion to the kinetic energy.
Quantum Problem .1 Angular Momentum
The various components of the angular quantity operator do not change, so it is not possible to measure two components at the same time. If we knew all three components of the angular momentum at the same time, this would mean that we could precisely measure the position and momentum of the particle at the same time, which would be a violation of the uncertainty principle.
