| course |
course_year |
question_number |
tags |
title |
year |
Quantum Mechanics |
IB |
61 |
IB |
2003 |
Quantum Mechanics |
|
1.II.18A |
2003 |
What is the significance of the expectation value
$$\langle Q\rangle=\int \psi^{*}(x) Q \psi(x) d x$$
of an observable $Q$ in the normalized state $\psi(x)$ ? Let $Q$ and $P$ be two observables. By considering the norm of $(Q+i \lambda P) \psi$ for real values of $\lambda$, show that
$$\left\langle Q^{2}\right\rangle\left\langle P^{2}\right\rangle \geqslant \frac{1}{4}|\langle[Q, P]\rangle|^{2}$$
The uncertainty $\Delta Q$ of $Q$ in the state $\psi(x)$ is defined as
$$(\Delta Q)^{2}=\left\langle(Q-\langle Q\rangle)^{2}\right\rangle .$$
Deduce the generalized uncertainty relation,
$$\Delta Q \Delta P \geqslant \frac{1}{2}|\langle[Q, P]\rangle| .$$
A particle of mass $m$ moves in one dimension under the influence of the potential $\frac{1}{2} m \omega^{2} x^{2}$. By considering the commutator $[x, p]$, show that the expectation value of the Hamiltonian satisfies
$$\langle H\rangle \geqslant \frac{1}{2} \hbar \omega .$$