Quantum mechanical integral equation problem

[bkmtkm]

Homework Statement

The question is;
for a qunatum mechanical particle,
Ψ(x) = [1/(a^1/2.π^1/4)].[e^-(x-x_o)2/2a].[e^ip₀x/h]

in here, x₀, p₀ and h are constants, so,

Homework Equations

what are the <x>; expetation value, and <P>;expectation value of momentum ?

The Attempt at a Solution

,
[/B]
first we have to simplified to integral, than maybe can be resolvable

 

[DrClaude]

first we have to simplified to integral, than maybe can be resolvable

So how can you express the expectation values in integral form?

 

[Apashanka]

The question is;
for a qunatum mechanical particle,
Ψ(x) = [1/(a1/2.π1/4)].[e-(x-xo)2/2a].[eip0x/h

for the expectation value of x ,use gamma function integral
e.g ∫xⁿexp(-ax^m)dx=
(1/m)(Γ(n+1)/n)/a^(n+1)/n

 

[Erik 05]

I did this with change of variable, y=x-x₀. gives integral of odd function = 0 plus standard integral (int exp(-y²/a) = sqrt(a)sqrt(pi)), result <x>=x₀/sqrt(a). p is a little more tricky, but uses similar integrals, and the imaginary terms eventually cancel out giving <p>=p₀/sqrt(a) - assuming that the 'constant h' is actually h bar.

 

[DrClaude]

I did this with change of variable, y=x-x₀. gives integral of odd function = 0 plus standard integral (int exp(-y²/a) = sqrt(a)sqrt(pi)), result <x>=x₀/sqrt(a). p is a little more tricky, but uses similar integrals, and the imaginary terms eventually cancel out giving <p>=p₀/sqrt(a) - assuming that the 'constant h' is actually h bar.

There is a little problem here, because the original wave function is not correctly normalised. I think that it should be $a^2$ in the first exponential.

 

[Erik 05]

I wondered that, similar examples I have seen have /2a², so <x>=x₀ and <p>=p₀