| course |
course_year |
question_number |
tags |
title |
year |
Quantum Mechanics |
IB |
45 |
IB |
2005 |
Quantum Mechanics |
|
2.II.16G |
2005 |
A particle of mass $m$ moving in a one-dimensional harmonic oscillator potential satisfies the Schrödinger equation
$$H \Psi(x, t)=i \hbar \frac{\partial}{\partial t} \Psi(x, t),$$
where the Hamiltonian is given by
$$H=-\frac{\hbar^{2}}{2 m} \frac{d^{2}}{d x^{2}}+\frac{1}{2} m \omega^{2} x^{2}$$
The operators $a$ and $a^{\dagger}$ are defined by
$$a=\frac{1}{\sqrt{2}}\left(\beta x+\frac{i}{\beta \hbar} p\right), \quad a^{\dagger}=\frac{1}{\sqrt{2}}\left(\beta x-\frac{i}{\beta \hbar} p\right)$$
where $\beta=\sqrt{m \omega / \hbar}$ and $p=-i \hbar \partial / \partial x$ is the usual momentum operator. Show that $\left[a, a^{\dagger}\right]=1$.
Express $x$ and $p$ in terms of $a$ and $a^{\dagger}$ and, hence or otherwise, show that $H$ can be written in the form
$$H=\left(a^{\dagger} a+\frac{1}{2}\right) \hbar \omega$$
Show, for an arbitrary wave function $\Psi$, that $\int d x \Psi^{*} H \Psi \geq \frac{1}{2} \hbar \omega$ and hence that the energy of any state satisfies the bound
$$E \geq \frac{1}{2} \hbar \omega$$
Hence, or otherwise, show that the ground state wave function satisfies $a \Psi_{0}=0$ and that its energy is given by
$$E_{0}=\frac{1}{2} \hbar \omega$$
By considering $H$ acting on $a^{\dagger} \Psi_{0},\left(a^{\dagger}\right)^{2} \Psi_{0}$, and so on, show that states of the form
$$\left(a^{\dagger}\right)^{n} \Psi_{0}$$
$(n>0)$ are also eigenstates and that their energies are given by $E_{n}=\left(n+\frac{1}{2}\right) \hbar \omega .$