Infinite square well, Probability of measurement of particle's energy

[machofan]

Homework Statement

Homework Equations

The Attempt at a Solution

I have managed to do the first 3 parts of the questions. The last two 4 markers are the ones I am having difficulties with. I have tried using the expansion postulate which states the wavefunction is equal to the sum of the expansion coefficent "a" and the eigenfunction. Then by squaring the expansion coefficient, this should provide the probability.

The wavefunction from part 3) was found to be 2Acos(kx), and I've tried integrating this by squaring it, but I notice that's not the right way to go about this problem.

Any help would be much appreciated.

 

[tman12321]

As you said, the wavefunction Φ may be expanded in terms of a linear combination of the ψ(n). The {ψ} form an orthonormal basis. What happens if you take <ψ(1)|Φ>?

 

[machofan]

As you said, the wavefunction Φ may be expanded in terms of a linear combination of the ψ(n). The {ψ} form an orthonormal basis. What happens if you take <ψ(1)|Φ>?

If you take ψ and Φ, then integrate from a to -a, would that provide the expansion coefficient a_{n}? In which case, by squaring this coefficient, this would provide the probability?

 

[tman12321]

This is a basic fact in quantum mechanics. You should consult your textbook so that you understand this, because you are bound to see it over and over again.