# Dirac Notation question

## Homework Statement

A particle in a harmonic oscillator potential in the following state after a time t:

## | ψ(t) > = \frac{1}{\sqrt{2}} [e^{(-iE_0 t/\hbar)} |ψ_0> + e^{(-iE_1 t/\hbar)} |ψ_1> ] ##

I want to write an expression for ## <ψ(t)| \hat{x} | ψ(t) > ##.

## Homework Equations

The answer is meant to be:

## <ψ(t)| \hat{x} | ψ(t) > = \frac{1}{2} [ <ψ_0| \hat{x} | ψ_1> e^{-i(E_1 - E_0)t/\hbar)} + <ψ_1| \hat{x} | ψ_0> e^{-i(E_0 - E_1)t/\hbar)}] ##

## The Attempt at a Solution

[/B]
## <ψ(t)| \hat{x} | ψ(t) > = \int{ψ^{*}(t) \hat{x} ψ(t)}##
## = \int{\frac{1}{\sqrt{2}} [ e^{(iE_0 t/\hbar)} ψ^{*}_0 + e^{(iE_1 t/\hbar)} ψ^{*}_1}] \hat{x} \frac{1}{\sqrt{2}}[ e^{-(iE_0 t/\hbar)} ψ_0 + e^{-(iE_1 t/\hbar)} ψ_1] ##

## = \int{\frac{1}{2} [ e^{(iE_0 t/\hbar)} ψ^{*}_0 \hat{x} e^{-(iE_0 t/\hbar)} ψ_0 + e^{(iE_0 t/\hbar)} ψ^{*}_0 \hat{x} e^{-(iE_1 t/\hbar)} ψ_1 + e^{(iE_1 t/\hbar)} ψ^{*}_1} \hat{x} e^{-(iE_0 t/\hbar)} ψ_0 + e^{(iE_1 t/\hbar)} ψ^{*}_1} \hat{x} e^{-(iE_1 t/\hbar)} ψ_1 ##

## = \frac{1}{2} [<ψ_0| \hat{x} | ψ_0 > + e^{-i(E_1 - E_0)t/\hbar} <ψ_0| \hat{x} | ψ_1 > + e^{i(E_1 - E_0)t/\hbar} <ψ_1| \hat{x} | ψ_0 > + <ψ_1| \hat{x} | ψ_1> ]##

This is a different answer than it should be, where am I going wrong?

Orodruin
Staff Emeritus
Homework Helper
Gold Member
No, it is correct. You are just missing one step of simplification.

No, it is correct. You are just missing one step of simplification.

Do you say that ##<ψ_0| \hat{x} | ψ_0 >## and ##<ψ_1| \hat{x} | ψ_1 >## are expectation values of position, which for the simple harmonic oscillator, are zero?

Orodruin
Staff Emeritus