###1. Time Evolution
####1.1. Superposition for particle in a box
Suppose we have an infinitely deep potential well with the particle in a linear superposition, for example, with equal parts of the first and second states of the well (This superposition is normalized)
For this superposition, the probability density:
Note that this probability density has a part that is oscillating time at an angular frequency . Note also that the absolute energy origin does not matter here for this measurable quantity, only the energy difference matters.
For the probability density is same as the probability density where
And it’s the same fact for and .
However, when you add the amplitude up and take the probability of the sum, the equal superposition of the two oscillates at th angular frequency (just like we worked out).
####1.2. Superposition for the harmonic oscillator
Quite generally, if we make a linear combination of two energy eigenstates with energies and . The resulting probability distribution with oscillate at the (angular) frequency . So, if we have a superposition wave function
then the probability distribution will be
In harmonic oscillator, we take the first and second state as example, the probability density is
where the angular frequency .
####1.3. The coherent state
The coherent state for a harmonic oscillator of frequency is
and the are the harmonic oscillator eigenstates.
Incidentally, note that for the expansion coefficients
This is the Poisson distribution from statistics with mean and standard deviation .
####2.1. Group velocity
Consider two waves at different frequencies and and suppose that the wave velocity is the same independent of frequency. Then the corresponding wavevector magnitude is the same for both wave. If we take two such waves of equal amplitudes and add them together, then we get spatial beats — a “spatial envelope”. The envelope moves at the same speed as the wave.
Algebraically, for two waves at different frequencies, one at frequency and wavevector and one at frequency and wavevector . By using complex exponential waves, we got a total wave
The equation can be rewritten as
Note here, because and then and .
But suppose the wave velocity is different for different frequencies (e.g. suppose the higher frequency wave has a slower velocity). Then the “envelope velocity” which we will call the group velocity is not the same as the underlying wave velocity. More precisely, we define
as group velocity. And
as phase velocity.
####2.2. Group velocity for a free electron
The small dispersion in glass gives significant effects in long fibers. Large dispersions are found near absorption lines. In waveguides, different spatial forms (modes) propagate at different velocities dispersion from geometry or structure. Any structure whose physical properties such as refractive index change on a scale with a wavelength will also show string “structural” dispersion.
For a free electron where , in one direction the Schroedinger equation is
with solution and . This means that
The group velocity of a free electron is given by
The group velocity does gives us which corresponds with out classical ideas of velocity and kinetic energy. This suggests it is meaningful to think of the electron as moving at the group velocity.
Note that phase velocity does not give us this kind of relation with , we have , then we can deduce
####2.3. Electron wavepackets
We can construct a “wavepacket” by putting together a liner superposition of energy eigensolutions. For a free electron or a similar particle of mass , the individual eignsolutions are plane waves. For propagating in the direction, these are of the form
for some chosen value of , and hence of
One convenient and useful set of values and amplitudes to choose is a set of equally spaced values with Gaussian amplitudes or “weights” for
Here is the center of the distribution of values, is width parameter for the Gaussian function. Note that this gives a “pulse” that is also Gaussian in space.
As we let time evolve, by simply adding up the terms in our wavepackets sum at each time. We can see the wavepacket propagate moving to the right and getting wider. A wavepackets that increases in width as it propagates is said to be “dispersing”. It gets wider because the change in with is not even linear. The effect is known as group velocity dispersion.
###3. Measurement, Operators and Expectation Values
####3.1. Quantum-mechanical measurement
Suppose we take some (normalized) quantum mechanical wave function and expand it in some complete orthonormal set of spatial functions . At least if we allow the expansion coefficients to vary in time. We know we can always do this:
Then the fact that is normalized means that we know the answer for the normalization integral
Because of the orthogonality of the basis functions, only the terms with survive the integration. Because of the orthonormality of the basis functions, the result from any such term will simply be , hence we have
On measurement of a state, the system collapses into the -th eigenstate of the quantity being measured with probability:
In the expansion of the state in the eigenfunctions of the quantity being measured is the expansion coefficient of the -th eigenfunction.
Suppose we do an experiment to measure the energy of some quantum mechanical system. We could repeat the experiment many times and get statistical distribution of results. Given the probabilities of getting a specific energy eigenstate, with energy . We would get an average answer:
where we call this average the “expectation value”.
In the following context, we are describing Stern-Gerlach experiment with electrons which has shocking different result than classical view of the experiment. The apparatus has a non-uniform magnetic filed where locally the magnetic filed is more concentrated near the north pole magnet face. Imagine firing some small magnets. Initially along the dashed line (you need to see the video to get this). Because the magnetic field is non-uniform, a vertical magnet will be deflected up or down. A horizontally-oriented magnet will not be deflected and magnets of other orientations should be deflected by intermediate amounts. After “firing” many randomly oriented magnets, we should end up with a line on the screen.
Electrons have a quantum mechanical property called spin. It gives them a “magnetic moment” just like a small magnet. And when we fire electrons with no particular “orientation” of the spin into the Stern-Gerlach apparatus, we got only two dots! The explanation of this would be that we are measuring the vertical component of the spin. There are two eigenstates of this component: “up” and “down”, so we have collapsed to the eigenstates.
####3.2. Expectation values and operators
In classical mechanics, the Hamiltonian is a function of position and momentum representing the total energy of the system. In quantum mechanical systems that can be analyzed by Schroedinger equation, we can define the entity
we can write the Schroedinger equation as
The entity is an “operator”. The most general definition of an operator is an entity that turns one function into another. The particular operator is called the Hamiltonian operator. Just like the classical Hamiltonian function, it is related to the total energy of the system. This Hamiltonian idea extends beyond the specific Schroedinger-equation definition we have so far which is for single, non-magnetic particles. In general, in non-relativistic quantum mechanics, the Hamiltonian is the operator related to the total energy of the system.
In order to understand the relation between the Hamiltonian operator, the wave function and the expectation value of the energy. We start by looking at the integral
where is the wave function of some system of interest. Let expend the wave function in the (normalized) energy eigenstates
Hence the integral becomes
Because of the orthonormality of the basis functions , the only terms in the double sum that survive are the ones for which . So
####3.3. Time evolution and the Hamiltonian
Taking Schroedinger’s time dependent equation and rewriting as
And now, we are targeting to seek the relationship between two different states in different times.
First we note that, because is a linear operator, for any number
Since this works for any function , we can write as a shorthand
Specifically, for example, for the energy eigenfunction
We can proceed inductively to define all higher powers: which will give, for an energy eigenfunction
Suppose the wave function at time is which we expand in the energy eigenfunction as , and we know that
We can write exponentials as power series,
Because we showed that ,
Because the operator and all its power commute with scalar quantities (numbers) we can rewrite
And this equation can be reverse to exponential