- #1
cleggy
- 29
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1. A particle in a harmonic potential energy well is in a state described by the initial wave function
Ψ(x, 0) = (1/√2) (ψ1(x)+ iψ3(x))
where ψ1(x)and ψ3(x) are real normalized energy eigenfunctions of the harmonic oscillator with quantum numbers n =1 and n = 3 respectively.
(a)
Write down an expression for Ψ(x, t) that is valid for all t> 0. Express your answer in terms of ψ1(x), ψ3(x)and ω0, the classical angular frequency of the oscillator.
(b)
Find an expression for the probability density function at any time t> 0. Express your answer in terms of ψ1(x), ψ3(x), ω0 and t.Use the symmetry of this function to show that the expectation value,<x> = 0 at all times.
I have reached Ψ(x, t) = (1/√2) (ψ1(x)exp(-3iw[tex]_{}0[/tex]t/2+ iψ3(x)exp(-7iw[tex]_{}0[/tex]t/2)
for part (b) I'm not sure how to calculate the probability density function at any time t>0 ?
I know that the probability density needs to be even function of x and so therefore being symmetrical about the centre of the well at all times.
Then using the sandwich integral to calculate <x>, it will yield zero as the integrand is an odd function.
Ψ(x, 0) = (1/√2) (ψ1(x)+ iψ3(x))
where ψ1(x)and ψ3(x) are real normalized energy eigenfunctions of the harmonic oscillator with quantum numbers n =1 and n = 3 respectively.
(a)
Write down an expression for Ψ(x, t) that is valid for all t> 0. Express your answer in terms of ψ1(x), ψ3(x)and ω0, the classical angular frequency of the oscillator.
(b)
Find an expression for the probability density function at any time t> 0. Express your answer in terms of ψ1(x), ψ3(x), ω0 and t.Use the symmetry of this function to show that the expectation value,<x> = 0 at all times.
Homework Equations
The Attempt at a Solution
I have reached Ψ(x, t) = (1/√2) (ψ1(x)exp(-3iw[tex]_{}0[/tex]t/2+ iψ3(x)exp(-7iw[tex]_{}0[/tex]t/2)
for part (b) I'm not sure how to calculate the probability density function at any time t>0 ?
I know that the probability density needs to be even function of x and so therefore being symmetrical about the centre of the well at all times.
Then using the sandwich integral to calculate <x>, it will yield zero as the integrand is an odd function.