QM Problems with wavefunction given in sin and cos.

[darkar]

Well, the question goes like this,

A particle of mass m is trapped in an infinitely deep one-dimensional potential well between x = 0 and x = a and at a time t=0,, the wave fuction is given as

Φ(x,t=0)=sin(((πx)/a))cos(((2πx)/a))

(i) What possible values may be found for energy of particle?
(ii) What is the expected (or average) value of the energy of the particle?

Well, we got the hints of using
sinAcosA = (1/2){sin(A+B)+sin(A-B)}

We got (1/2){sin(3*pi*x/a) - sin(pi*x/a)}... but not sure what to do next.

 

[Physics Monkey]

Those sure look like energy eigenstates to me.

 

[topsquark]

Well, the question goes like this,

A particle of mass m is trapped in an infinitely deep one-dimensional potential well between x = 0 and x = a and at a time t=0,, the wave fuction is given as

Φ(x,t=0)=sin(((πx)/a))cos(((2πx)/a))

(i) What possible values may be found for energy of particle?
(ii) What is the expected (or average) value of the energy of the particle?

Well, we got the hints of using
sinAcosA = (1/2){sin(A+B)+sin(A-B)}

We got (1/2){sin(3*pi*x/a) - sin(pi*x/a)}... but not sure what to do next.

What is the definition of an expectation value? (Hint: It's involves an integral over the state of the system.)

-Dan

 

[darkar]

The definition of expected value shd be integration of f(x) * psi^2 dx. Am i rite?

Well, the biggest problem is how shd we start, what can u say abt a wave function with (1/2){sin(A+B)+sin(A-B)}?

 

[cepheid]

We got (1/2){sin(3*pi*x/a) - sin(pi*x/a)}... but not sure what to do next.

And in DIRECT response to the above comment:

Those sure look like energy eigenstates to me.

Does that make it a bit more obvious? I didn't bother checking to see if you did your trig right, but if you did, then you have a linear combination of eigenfunctions for the infinite square well problem. That should make it pretty clear how to proceed.

 

[darkar]

Umm, well, there's another thing, if u do integration of square of the given psi^2, u get a/4 instead of 1, shouldn't it be 1?

 

[darkar]

Umm, all rite, got it , i need to normalize the equation.

Thanks for the hints everyone ~