# 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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