2.6. Problems

2.1.

The product of the exponentials of two non-commuting operators $X$ and $Y$ can be written as $$e^X e^Y = \exp(X+Y+[X,Y]/2+\ldots)$$ This is the so-called Campbell Baker Hausdorff formula (or CBH formula). The aim of this problem is to derive this formula.

1. First, show that $\log(1+x) \approx x - x^2/2 + x^3/3 - \ldots$.

2. Now expand the formula $\log(e^X e^Y)$ to second order in $X$ and $Y$ to derive the CBH formula.

3. If you're brave, you may try to find the third order expansion to find the next term.

2.2.

We consider a particle moving in one dimension. The state of the particle is $\left| \psi\right\rangle$.

1. Show that $$\left\langle x+a | \psi \right\rangle = \left\langle x \right| e^{{\rm i}a \hat{p}/\hslash} \left| \psi \right\rangle$$ by inserting a unit operator using completeness of the momentum basis $\left| p \right\rangle$.

2. Demonstrate the same result by writing $\hat{p}$ as the operator $-{\rm i}\hslash d/dx$ and Taylor expanding $\psi$ around $x$.

3. Show that for a wave function in three dimensions: $$\left\langle {\bf r}' | \psi \right\rangle = \left\langle {\bf r} \right| e^{{\rm i}\alpha \hat{L}_z/\hslash} \left| \psi \right\rangle,$$ where ${\bf r}'$ is the vector ${\bf r}$ rotated about an angle $\alpha$ around the $z$-axis.

4. (The following two parts were already addressed in problem 1.9). Show that the operator $\exp({\rm i}\alpha \sigma_z/2)$ rotates a spin-1/2 state about an angle $\alpha$ around the $z$-axis.

5. Give the rotation operator for the wave function describing a spin-1/2 particle.

2.3.

The Lagrangian of a harmonic oscillator is $L=(m/2) (\dot{x}^2-\omega^2x^2)$. Show that the classical action is: $$S_{\text{cl}}=\frac{m\omega}{2\sin{\omega T}}\left[(x_{\text{i}}^2+x_{\text{f}}^2)\cos{\omega T}-2x_{\text{i}}x_{\text{f}}\right],$$ where $T=t_{\text{f}}-t_{\text{i}}$

Hint: the definition of the action is: $S=\int_{t_{\text{i}}}^{t_{\text{f}}} \! L(\dot{x},x,t) \, dt$. Make use of a classical trajectory of the harmonic oscillator: $x(t)=A\cos{\omega t}+B\sin{\omega t}$, which satisfies boundary conditions: $x(t_{\text{i}})=x_{\text{i}}$ and $x(t_{\text{f}})=x_{\text{f}}$.

2.4.

The path integral formalism expresses the probability to move from a point $x_{\text{i}}$ at time $t_{\text{i}}$ to a point $x_{\text{f}}$ at time $t_{\text{f}}$ in terms of a sum over all paths: $$K(x_{\text{f}}, t_{\text{f}}; x_{\text{i}}, t_{\text{i}}) = \sum_{\text{Paths}} \exp({\rm i}S_{\text{path}}/\hslash).$$ We write the path as a sum of the classical path $x_{\text{class}}(t)$ and a fluctuation: $$x(t)=x_{\text{class}}(t) + \delta x(t),$$ where $\delta x(t_{\text{i}}) = \delta x(t_{\text{f}}) = 0$, since the positions at the beginning and at the end of the path are fixed.

We consider a free particle.

1. Give the action for the classical path for a free particle (that is, $V(x)=0$ for all $x$).

2. Show that the action for a general path can be written as $$S = S\left[x_{\text{class}}\right] + \frac{m}{2} \int \delta \dot{x}^2(t) \; dt.$$ Also show that $$K(x_{\text{f}}, t_{\text{f}}; x_{\text{i}}, t_{\text{i}}) = K(0, t_{\text{f}}; 0, t_{\text{i}}) \exp\left[ \frac{{\rm i}}{\hslash} S(x_{\text{class}})\right].$$

3. The properly normalised form of the discretised path integral is given by $$K(0, t_{\text{f}}; 0, t_{\text{i}}) = \lim_{N\rightarrow \infty} \left( \frac{m}{2\pi{\rm i}\epsilon\hslash} \right)^{(N+1)/2} \int \prod_{n=1}^{N+1} dx_n \exp\left[ \frac{{\rm i}}{\hslash} \epsilon \sum_{n=1}^{N} L(x, \dot{x}) \right].$$ In the exponent, $\dot{x}$ means $(x_{n}-x_{n-1})/\epsilon$; $\epsilon = (t_{\text{f}}-t_{\text{i}})/(N+1)$. Furthermore, $x_0 \equiv x_{\text{i}}$ and $x_{N+1} \equiv x_{\text{f}}$.

   The integration over all possible paths $\delta x$ can be performed in the discretised path integral: $$I_N = \int \prod_{i=1}^{N+1} dx_n \; \exp\left( {\rm i}\frac{\epsilon}{\hslash} \sum_{n=1}^{N} \frac{m \dot{x}^2}{2} \right).$$ It can be shown (using e.g. induction) that $$I_N = \frac{1}{\sqrt{N+1}} \left( \frac{2\pi {\rm i}\epsilon\hslash}{m} \right)^{N/2}.$$ If you're in for a challenge, you may try to prove this, but that is not required here.

   Show, using this result, that $$K(x_{\text{f}}, t_{\text{f}}; x_{\text{i}}, t_{\text{i}}) = \sqrt\frac{m \hslash}{2\pi {\rm i}(t_{\text{f}} - t_{\text{i}})} \exp\left( \frac{{\rm i}m}{2\hslash} \frac{(x_{\text{f}}-x_{\text{i}})^2}{(t_{\text{f}}-t_{\text{i}})}\right).$$

4. Find this propagator directly, using $$K(x_{\text{f}}, t_{\text{f}}; x_{\text{i}}, t_{\text{i}}) = \left\langle x_{\text{f}} \right| e^{-{\rm i}(t_{\text{f}}-t_{\text{i}}) p^2/2m\hslash} \left|x_{\text{i}} \right\rangle.$$ Hint: insert two unit operators in this expression, formulated as integrals over $p$.