Quantum Mechanics

The main topics here are the mathematical formulation of Hilbert spaces, understanding the Schrödinger equation for bound states and investigating scattering processes.

Mathematical Foundation

Hilbert spaces, operators, bras and kets.

Hermitean Operators Have Real Eigenvalues

A fundamental axiom of quantum mechanics states that observables are represented by hermitean operators acting in a Hilbert space. The reason behind this is that the result of a measurement should yield a real number instead of a complex one.

Show that the eigenvalues of a hermitean operator $A$ are real!

For a hermitean operator, $A=A^\dagger$.

Operator Exponentials

Consider a two-dimensional Hilbert space with basis vectors $|b_i\rangle$ ($i=1,2$). A hermitean operator is given by $B = |b_1\rangle\!\langle b_2| + |b_2\rangle\!\langle b_1|$.

1. Calculate the operator $\exp(\text i \epsilon B)$ for $\epsilon\in\mathbb R$!

Use a Taylor series.

2. Calculate $\exp(\text i \epsilon B)|b_1\rangle$!

Operator Derivatives

The operators $A(\lambda)$ and $B(\lambda)$ depend on a real parameter $\lambda$, whereas the operator $C$ is independent of $\lambda$. Prove the following statements!

1. $\frac{\text d}{\text d\lambda} (AB) = \frac{\text dA}{\text d\lambda}B + A\frac{\text dB}{\text d\lambda}$

2. $\frac{\text d}{\text d\lambda}A^{-1} = -A^{-1}\frac{\text dA}{\text d\lambda}A^{-1}$

3. $\frac{\text d}{\text d\lambda}\text e^{\lambda C} = C \text e^{\lambda C}$

4. $\frac{\text d}{\text d\lambda}A^n = \sum\limits_{k=1}^n A^{k-1} \frac{\text dA}{\text d\lambda}A^{n-k}$

Baker-Campbell-Haussdorf (Simplified)

If the commutator of $[A,B]$ with the two operators $A$ and $B$ vanishes, that is $[A,[A,B]]=0$ and $[B,[A,B]]=0$, a simplified version of the Baker-Campbell-Haussdorf formula holds: $$\text e^A \,\text e^B = \text e^{A+B+\tfrac{1}{2}[A,B]}.$$

1. Prove this statement!

Construct a differential equation for the operator $\text e^{\lambda A}\text e^{\lambda B}$ in $\lambda$ and integrate it.

2. Check that this equation can be applied for $A=\hat x$ and $B=\hat p$ and prove the following statement: $$\text e^{-\frac{\text i}{\hbar}ap}\,\text e^{\text ibx}\,\text e^{\frac{\text i}{\hbar}ap} = \text e^{\text i b(x-a)}, \quad a,b\in\mathbb R.$$Note the connection to the translation operator $T(a)$!

3. Check that this equation can be applied for the creation and annihilation operators of the harmonic oscillator, $A=\hat a$ and $B=\hat a^\dagger$, and prove the following statement: $$\text e^{ca^\dagger-c^* a} = \text e^{-\tfrac{1}{2}|c|^2}\,\text e^{ca^\dagger}\,\text e^{-c^* a} = \text e^{\tfrac{1}{2}|c|^2}\,\text e^{-ca^\dagger}\,\text e^{c^* a}, \quad c\in\mathbb C.$$

Proving Bloch's Theorem

Bloch's theorem states that for particles in a perfect crystal of lattice size $a$, there is a basis of energy eigenstates that can be written as, $$\psi(\textbf r)=\text e^{\text i\textbf {kr}}u(\textbf r),$$where $u(\textbf r)$ has the same periodicity as the lattice: $u(x+a)=u(x)$. This is called a Bloch wave.

Verify Bloch's theorem via the eigenvalue equation of the translation operator $T(a)$

The translation operator is defined as $T(\Delta\textbf x)=\text e^{-\text i \Delta\textbf{x}\cdot \textbf{p}/\hbar}$.

Parity Operator

For a one-dimensional system, the parity operator $\Pi$ acts on the state $|x\rangle$ as $$\Pi|x\rangle = |-\!x\rangle,$$where the state is defined via $\hat x|x\rangle = x|x\rangle$.

Prove the following statements!

1. $\Pi = \Pi ^\dagger = \Pi^{-1}$, therefore its eigenvalues are $\pm 1$.

2. If the potential obeys $V(-x)=V(x)$, the eigenfunctions of the Hamiltonian are either even or odd functions in $x$.

Show that the eigenvalue of the parity operator (that is, whether a function is even or odd) is a conserved quantity.

Commutator Relations

The commutator of the position operator $x$ and the momentum operator $p$ in one dimension is given by, $$[x,p]=\text{i}\hbar.$$

Prove the following equations!

1. $[x,p^n] = \text{i} \hbar \frac{\partial}{\partial p}(p^n)$.

2. $[x,f(p)] = \text{i}\hbar \frac{\partial f(p)}{\partial p}$

Wave Functions

Normalization, probability densities.

One-dimensional Wave function

The wave function of a particle is given by, $$\psi(x) = N\, \text e^{-a |x|} \cos (x)\,\text e^{\text i \varphi x},\quad -\infty <x<\infty,$$ where $a>0$ and $\varphi\in\mathbb R$.

1. For a given $a$ and $\varphi$, calculate $N$ such that the wave function is normalized!

Normalizaion condition: $\int_{-\infty}^\infty \psi^* \psi \stackrel{!}= 1$.

2. What is the probability of finding the particle in the interval $[-\tfrac{\pi}{2},\tfrac{\pi}{2}]$?

Integrate the probability density over a suitable interval.

Time-Independent Schrödinger Equation

Bound states, eigenfunctions and eigenenergies.

The Time-Independent Schrödinger Equation

The time-dependent Schrödinger equation in one dimension is given by $$H\psi(x,t) = -\text i\hbar\frac{\text d}{\text dt}\psi(x,t),$$ where $H$ is the Hamiltonian of the system.

Using an ansatz for the wave function $\psi(x,t)$, show that the time-independent Schrödinger equation is given by $H\psi(x) = E\psi(x)$!

Use a separating ansatz like $\psi(x,t)=\psi(x)f(t)$ with a suitable function $f(t)$.

Free Particle (1D)

The Schrödinger equation for a free particle in one dimension is given by $H\psi = E\psi$, where $H$ is the free Hamiltonian, $$H_\text{free} = -\frac{\hbar^2}{2m}\frac{\text d^2}{\text dx^2}.$$

Using an ansatz for the wave function $\psi(x)$, solve the Schrödinger equation and calculate the eigen energies!

A suitable ansatz would be $\psi(x) = A\, \text e^{\text i kx}$ with $A,k\in\mathbb R$.

Particle with a Constant Potential (1D)

A particle of mass $m$ is travelling in a constant potential $V(x)=V_0$.

Solve the one-dimensional Schrödinger equation to get the wave function $\psi(x)$ and the eigen energies $E$ ($E>V_0$)!

This case is very similar to the one without any potential.

Infinite Square Well (1D)

A particle of mass $m$ is trapped in an infinite potential well, $$V(x)=\begin{cases}0 & |x|\le a\\+\infty & |x|>a\end{cases}.$$

Find the bound-state wave functions and their energies (Don't forget to normalize the wave functions)!

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Finite Square Well (1D)

A particle of mass $m$ is trapped in a finite potential well, $$V(x)=\begin{cases}-V_0 & |x|\le a\\0 & |x|>a\end{cases}.$$

Find the bound-state wave functions and their energies (Assume $-V_0 < E < 0$ and don't forget to normalize the wave functions)! What is the smallest value of $V_0$ that allows a bound state for any given $a$?

Delta Potential (1D) - First look

In the neighborhood of a delta potential, the wave function is not smooth anymore.

For the potential $$V(x) = -\Lambda\, \delta(x-x_0)$$ find the boundary condition for the first derivative of $\psi(x)$ when it approaches $x_0$ from the right and from the left!

Integrate the Schrödinger equation along a small region near $x_0$.

Delta Potential (1D)

A particle of mass $m$ encounters a delta potential: $$V(x) = -\Lambda\, \delta(x), \quad \Lambda >0.$$

Find the bound states of this system and the corresponding energies!

Infinite Square Well (2D)

A particle of mass $m$ is trapped in a two-dimensional infinite square well potential, $$V(x,y) = V_x(x)+V_y(y),$$$$V_x(x) = \begin{cases}0 & |x|\le a\\+\infty & |x|>a\end{cases}, \quad V_y(y) = \begin{cases}0 & |y|\le b\\+\infty & |y|>b\end{cases}.$$

Find the bound states of this system and the corresponding energies! Are there degenerate energies in this system?

A suitable ansatz for the wavefunction could be $\psi(x,y) = X(x)Y(y)$.

Infinite Square Well (3D)

A particle of mass $m$ is trapped in a three-dimensional infinite square well potential, $$V(x,y,z) = V_x(x)+V_y(y)+V_z(z),$$$$V_x(x) = \begin{cases}0 & |x|\le a\\+\infty & |x|>a\end{cases}, \quad V_y(y) = \begin{cases}0 & |y|\le b\\+\infty & |y|>b\end{cases}.$$ $$V_z(z) = \begin{cases}0 & |z|\le c\\+\infty & |z|>c\end{cases}.$$

Find the bound states of this system and the corresponding energies! Are there degenerate energies in this system?

A suitable ansatz for the wavefunction could be $\psi(x,y,z) = X(x)Y(y)Z(z)$.

Dirac Comb (1D)

A particle of mass $m$ is trapped in a one-dimensional, periodic potential, $$V(x) = -\frac{\hbar^2}{m} \sum\limits_{n=-\infty}^\infty \delta(x+na),\quad n\in\mathbb N, a\in\mathbb R$$

Find the eigenvalues of the system and show that there are allowed and forbidden values for the energy!

A suitable ansatz for the wave function is a Bloch wave: $$\psi_\text{Bloch}(x) = \text e^{\text ikx}\,u(x),$$ where $u(x)$ is a periodic function with respect to the potential: $u(x+a) = u(x)$, and $k$ is a parameter for which $-\frac{\pi}{a}\le k\le \frac{\pi}{a}$.

Potential Barriers

One-dimensional scattering, Transmission, Reflection.

Single Barrier

A particle of mass $m$ is propagating along the $x$-axis towards a barrier. The potential is given by $$V(x) = \begin{cases}0 & x< 0\\ V_0 & 0<x< a \\ 0 & x> a\end{cases}.$$

1. Make an ansatz for the wave function in the three regions $x<0$, $0<x<a$ and $x>a$!

2. Use the boundary conditions at $x=0$ and $x=a$!

3. Find the probability for transmission and reflection for this system and check that $R+T=1$!

Delta Barrier

A particle of mass $m$ is propagating along the $x$-axis towards a potential barrier. The potential is given by $$V(x) = -\Lambda\, \delta(x),\quad \Lambda >0.$$

1. Make an ansatz for the wave function in the two regions $x<0$ and $x>a$!

2. Use the boundary conditions at $x=0$!

3. Find the probability for transmission and reflection for this system and check that $R+T=1$!

4. What changes for $\Lambda < 0$?

Measurement

Expectaton values, uncertainties, Heisenberg's uncertainty.

Infinite Square Well - Energy

A particle of mass $m$ is trapped in an infinite potential well, $$V(x)=\begin{cases}0 & |x|\le a\\+\infty & |x|>a\end{cases}.$$At time $t=0$, the particle is in the state $$\psi(x,t=0)=N(a^2-x^2),\quad x\in[-a,a].$$

1. For $t=0$, calculate the probability $P_n$ for finding the particle in the $n$-th eigenfunction of the infinite square well!

2. Calculate the expectation value of the energy, i.e. the Hamiltonian $\langle H\rangle$ as well as the uncertainty in the energy $\Delta E$!

The uncertainty of a quantity is given by $\Delta A = \sqrt{\langle A^2\rangle - \langle A\rangle^2}$.

Infinite Square Well - Position and momentum

A particle of mass $m$ is trapped in an infinite potential well, $$V(x)=\begin{cases}0 & |x|\le a\\+\infty & |x|>a\end{cases}.$$At time $t=0$, the particle is in the state $$\psi(x,t=0)=N(a^2-x^2),\quad x\in[-a,a].$$

1. Calculate the following expectation values: $\langle x\rangle$, $\langle x^2\rangle$, $\langle p\rangle$ and $\langle p^2\rangle$!

Use the momentum operator $\hat p = -\text i\hbar \frac{\text d}{\text dx}$.

2. Calculate the uncertainties $\Delta x$ and $\Delta p$ and compare their results to Heisenberg's uncertainty relation!

The uncertainty of a quantity is given by $\Delta A = \sqrt{\langle A^2\rangle - \langle A\rangle^2}$.
Heisenberg's uncertainty relation is given by $\Delta x \cdot \Delta p \ge \frac{\hbar}{2}$.

Momentum of an Eigenfunction

Show that the expectation value of a particle's momentum, $\langle p\rangle$, is zero if that particle is in an eigenfunction of a general Hamiltonian $H=\frac{p^2}{2m}+V(x)$!

Show that the momentum operator can be written as $\hat p = \frac{\text im}{\hbar}[\hat H, \hat x]$ and use this expression in order to calculate the expectation value.

Virial Theorem

The Hamiltonian of a system is given by $$H=T+V=\frac{p^2}{2m}+V(x),$$ where the potential is a homogenous function of degree $n$: $V(\lambda x) = \lambda^n V(x)$.

Assume the system is in an eigenfunction $\psi$ of the Hamiltonian. Show that the expectation values of kinetic energy and potential energy are related via $$\langle T\rangle = \frac{n}{2}\langle V\rangle,$$where $\langle T\rangle = \langle \psi|T|\psi\rangle$!

Invesitgate the expectation value of the commutator of the Hamiltonian with the observable $A:=\frac{1}{2}(xp+px)$.

Harmonic Oscillator

Ladder operators, coherent states.

Parity Operator

For a one-dimensional system, the parity operator $\Pi$ acts on the state $|x\rangle$ as $$\Pi|x\rangle = |-\!x\rangle,$$where the state is defined via $\hat x|x\rangle = x|x\rangle$.

Show that the parity operator for the harmonic oscillator can be written as $\Pi = \text e^{\text i\pi \, a^\dagger a}$!

Coherent States

Coherent states in the quantum harmonic oscillator are defined via the eigenvalue equation of the annihilation operator: $a|\alpha\rangle = \alpha |\alpha\rangle$, where $\alpha\in\mathbb C$.

1. Show that $|\alpha\rangle = \text e^{-|\alpha|^2/2}\,\text e^{\alpha a^\dagger}|0\rangle = \text e^{-|\alpha|^2/2}\, \sum\limits_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}|n\rangle$!

2. Calculate the uncertainties of position $\Delta x$ and momentum $\Delta p$ and show that coherent states fulfil the equality in Heisenberg's uncertainty relation!

3. Calculate the energy uncertainty $\Delta E$ of a coherent state!

2D Harmonic Oscillator - Eigenvalues

The potential of an isotropic, two-dimensional harmonic oscillator is given by $V(x,y) = \tfrac{1}{2}m\omega^2(x^2+y^2)$.

1. Calculate the energy eigenvalues via the creation and annihilation operators $a_x$, $a_x^\dagger$, $a_y$ and $a_y^\dagger$ that are constructed via $\{x,y,p_x,p_y\}$.

2. Calculate the energy eigenvalues via the creation and annihilation operators for right- and left-circular quanta, defined by: $$a_R = \frac{1}{\sqrt 2}(a_x-\text ia_y),\quad a_L = \frac{1}{\sqrt 2}(a_x+\text ia_y).$$

3. Show that the eigenfunctions of b. are eigenfunctions of the $z$-component of the angular momentum $L_z = xp_y-yp_x$.

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