Quantum Mechanics

Bloch’s theorem

The eigenfunctions of the Schrödinger equation for a periodic potential, V(x + a) = V(x), are the product of a plane wave, e^{ikx} times a function u(x). ψ(x) = u(x)e^{ikx}where u(x) has the period of the lattice, i.e., u(x) = u(x + a)

Compatible operators

Two physical observables are said to be compatible if the operators representing them have a common set of eigenfunctions. Compatible operators commute.

Hence, a measurement of one operator will place the system into an eigenvalue of that operator, and then a subsequent measurement of the other operator will have a completely predictable result.

Degenerate eigenfunctions

If two or more eigenfunctions share the same eigenvalue then they are said to be degenerate. Any linear combination of degenerate eigenfunctions is also an eigenfunction with the same eigenvalue.

Particle flux

Represents the average number of particles passing a point x per unit time t.

Postulate 1

For a given system, a wavefunction exists that contains all information about the system.

• The wave function is a continuous, square-integrable, single-valued function of the parameters of all the particles and of time.

Postulate 2

Physical observable quantities, such as position, momentum, or energy, may be extracted from the wavefunction using a mathematical operator. The relevant equation is the eigenvalue equation \hat Q\phi_n=q_n\phi_n, where \hat Q is the operator, \phi_n are the eigenfunctions, and q_n are the eigenvalues that correspond to the quantities that can be measured in the laboratory. We impose a mathematical requirement that the operators are linear Hermitian operators, which ensure real eigenvalues. Critically, after a measurement is made, the wavefunction must be an eigenfunction of the system.

Postulate 3

The form of the position operator is \hat R=\vec r and the momentum operator is \hat P=-i\hslash\vec \nabla and using these we can construct other quantum mechanical operators using the functional relationship of their corresponding classical functions.

Postulate 4

We can represent any general wavefunction for a given system as a linear combination of the eigenfunctions of the system, \psi=\sum_na_n\phi_n. Moreover, the quantity |a_n|² tells us the probability that a measurement of the wavefunction will yield the result q_n.

Postulate 5

The time evolution of the wavefunction when it remains undisturbed is governed by the TDSE i\hslash\frac{\partial\psi(\vec r, t)}{\partial t}=\hat H \psi(\vec r, t). For systems where the potential is independent of time, the general time dependent eigenfunctions are given by \psi(\vec r, t)=\sum_na_n\phi_n(\vec r)e^{-iE_nt/\hslash}

Born rule

The quantity |\psi(x,t)|² represents the probability density for determining the position of a particle at a time t. For this to be a valid probability density, the total probability of finding the particle somewhere in space must be unity, so \int_{-\infty}^\infty|\psi(x,t)|²\,\mathrm{d}x=1.

Hermitian operator

An operator \hat Q is Hermitian if the following condition is true: \int_{-\infty}^{\infty} f^*(x)\, \hat Q\, g(x) \,\,d x=\int_{-\infty}^{\infty} g(x) \left(\hat Q f(x) \right)^* d x, where f(x)\rightarrow 0 and g(x)\rightarrow 0 as |x|\rightarrow\infty.

Linear operator

An operator \hat Q is linear if, for any two functions f₁ and f₂ and constants c₁ and c₂:\hat Q(c_1f_1+c_2f_2)=c_1\, \hat Q(f_1)+c_2\, \hat Q(f_2)where the constants c1 and c₂ may be complex.

Require operators that represent physical observables to be linear so that we can construct superpositions of quantum states.

Commutation relations of \hat X and \hat P

[{\hat{X}}_{i},{\hat{X}_{j}}]=0,\quad[{\hat{P}_{i}},{\hat{P}_{j}}]=0,\quad[{\hat{P}_{i}},{\hat{X}_{j}}]=-i\hslash\,\delta_{ij}

Expectation value

The average set of repeated measurements on identical systems. \langle\hat{Q}\rangle=\int_{-\infty}^{\infty}\psi^{*}\hat{Q}\psi\,dx=\sum_{n}\left|a_{n}\right|^2q_{n}

Stationary state

A wavefunction \psi(x,t) is time-independent if it corresponds to a single energy eigenstate of the system, so there is no interference between different time-dependence of states of different energy in |\psi(x,t)|².

That is, although the wavefunctions of a stationary state wavefunction depend on time, their probability density does not.

Why we require quantum operators representing physical observables to be Hermitian

Hermitian operators ensure real eigenvalues and orthogonal eigenvectors.

How the free particle solution is modified by the periodic potential (Kronig-Penney model)

The free electron dispersion is modified by the opening of band gaps. These gaps open up at values of wave number satisfying ka = n\pi where a is the spacing between barriers in the potential.

Electronic properties of a semiconductor vs a metal

Given by Kronig-Penney model.

• In the case of a metal the highest filled state lies in the middle of a band, while in a semiconductor the highest filled state lies at the top of a band.

• Metals are therefore electrically conductive because a small applied bias can raise the energy of an electron to an unoccupied state.

• In the case of a semiconductor, electrons near the top of the band must be given an energy at least equal to the band gap.

Principle quantum number n

Determines the total energy, n=1,2,3…

Orbital angular momentum quantum number l

Determines the magnitude of angular momentum, l=0,1,2,…n-1

Magnetic quantum number m

Determines the projection of angular momentum onto z-axis, m=-l,…,-1,0,1,…l

Orthogonality condition

Eigenfunctions \phi_n(x) are orthogonal if: \int_{-\infty}^\infty\phi_n(x)\phi_m(x)\,\mathrm{d}x=0 for n\neq m.

If n=m, then this integral equals 1.

Orthonormality condition is the same, except one of the eigenfunctions in the integral is its complex conjugate.

Stern-Gerlach experiment

• Result showed two groups of atoms deflected in opposite directions

• Deflection implies a force on the atoms proportional to the gradient of the magnetic field and orientation of magnetic moment

• Classically, would expect a continuous distribution as any orientation of the magnetic moment is possible

• From orbital angular momentum, would expect an odd number of deflections (and for s-state particles, with m=0, would expect no deflection)

• Therefore experimental result implies the electron has an intrinsic magnetic moment with two possible orientations corresponding the magnetic quantum spin number

Quantum tunnelling

There is a finite probability for a quantum particle to tunnel through a potential energy barrier that would be forbidden classically.

Probability of finding a particle

The probability of finding a particle in a region on the x-axis between x = a and x = b is P_{a,b}=\int_{a}^{b}\Psi^*\Psi\, dx

Net flux

The net flux passing through this region at a time t is given by\text{net flux}=Γ(a)-Γ(b), which is also equal to the rate of change of the probability density of finding the particle between points a and b, that is Γ(a)-Γ(b)=\frac{dP_{a,b}}{dt}

Uncertainty principle

The components of position, \vec r, and momentum, \vec p, cannot be known with absolute precision at the same time. \Delta x \Delta p_x \geq \frac{\hslash}2 where \Delta x and \Delta p_x represent uncertainties.

Complete eigenfunctions

It is always possible to expand a function in terms of the eigenfunctions of the Hermitian operator.