Normalised Energy Eigenfunction (Probability with Dirac Notation)

[Unto]

Homework Statement

Normalised energy eigenfunction for ground state of a harmonic oscillator in one dimension is:

〈x|n〉=α^(1/2)/π^(1/4) exp(-□(1/2) α^2 x^2)

n = 0

α^2=mω/h

suppose now that the oscillator is prepared in the state:

〈x|ψ〉=σ^(1/2)/π^(1/4) exp(-(1/2) σ^2 x^2)


What is the probability that a measurement of the energy gives the result E_0 = 1/2 hω?

Homework Equations

The Attempt at a Solution

I squared both states as that is the probability of the states, but I cannot see where the energy eigenvalue would fit into it.

It probably something glaringly obvious, but I haven't done this in a long time and certainly not with dirac notation. Please help.

 

[Unto]

I squared the states to get probabilities, but now I'm left with 2 probability amplitudes that I'm stumped on how to use.

What can I use the energy for?

I know n = 0, but do I have to find the expectation value of getting the energy?

 

[dextercioby]

Apply the definition. The probability should be equal to the square modulul of some complex number, namely the scalar product of the 2 states involved. Which are the 2 states involved ?

 

[Unto]

Since both states are real, their conjugates equal their normal counterparts. So I end up using an Identity relation to superimpose the states anyway (underhanded trick from my QM lecturer :p ).

Ok so I'm multiplying and integrating over unity, hopefully a useful co-efficient should pop out which I should then modulus square for the probability? Will show my working a hot second.

 

[Unto]

How do I integrate e^[(-1/2)x^2] dx from -∞ to ∞ ?

 

[dextercioby]

It's the typical gaussian integration. You should find its value in your QM book, or statistics book. Look it up.