QM eigenfns and momentum op

[fredrick08]

Homework Statement

which of the following functions are eigenfunctions of the momentum op? what are the eigenvalues in these cases?

$$\psi$$(x)=Asin(kx)
$$\psi$$(x)=Asin(kx)-Acos(kx)
$$\psi$$(x)=Acos(kx)+iAsin(kx)
$$\psi$$(x)=Aexp((ik(x-a))

Homework Equations

Pop=hbar/i(d/dx)

The Attempt at a Solution

ok now my understanding of an eigenfn is that if u multiply the operator and function, ad the result is the function*constant, then the that function is an eigenfn, I am not quite sure though what the eigen value is... i think its the value of the constant. so i did the math(its probably wrong)

1.(hbar/i)(Akcos(kx))$$\neq$$const*fn
2.(hbar/i)(Akcos(kx)+Aksin(kx))$$\neq$$const*fn
3.(hbar/i)(Aksin(kx)+iAkcos(kx))=(hbar/i)(k)*fn, thus eigen value=(hbar*k/i)?
4.(hbar/i)(ikAexp(ik(x-a)))=(hbar*k)*fn, thus eigen value=hbar*k

please can some help me if I am wrong... I am just a bit unsure, i it looks ok, but can anyone confirm, TY

 

[Thaakisfox]

The eigenvalue problem of an operator is:

$$A\psi=\lambda \psi$$

A is the operator, $$\lambda$$ is the eigenvalue, and $$\psi$$ is the eigenfunction.

your logic was correct, and you calculated the first two correctly and the last one correctly, but you didnt differentiate the third one correctly ;)

You could even guess the eigenvalues of the momentum operator, before calculating the above. What does the eigenvalue of these operators mean in QM ;))

 

[fredrick08]

ok thankyou very much, ok then to fix up the 3rd one is it...
(hbar/i)(-Aksin(kx)+iAkcos(kx))=(-hbar/i)(k)*fn thus eigen value is (-hbar*k/i)

does that seem better.. always forget bout negative... lol, is an eigen value allowed to be complex? a complex number is still constant right?

 

[Thaakisfox]

Thats still not right :(

According to what you wrote:

$$-A\sin(kx)+iA\cos(kx)$$

is the original function, but that is :$$A\cos(kx)+iA\sin(kx)$$

So you should write:

$$-A\sin(kx)+iA\cos(kx)=i\cdot i A\sin(kx)+iA\cos(kx)=i(A\cos(kx)+iA\sin(kx))$$

Hence you have to multiply the number you got by i to get the eigenvalue.

And the momentum operator, and all of the operators representing phyisical quantities in QM are hermitian or self adjoint, hence the always have real eigenvalues, so if you get a complex eigenvalue, then you screwed something

 

[fredrick08]

ahh ok yes i thought so, to confirm again the correct answer should be
(hbar/i)(k)(i(Acos(kx)+iAsin(kx)))=hbar*k*fn, so hbar*k is eigenfn... that makes sense because its the same fn as the 4th one... i think.

can i also ask, how do the type ur calculations using latex... i have no idea how, and it would help a lot = )

 

[Pengwuino]

Clicking on the actual formulas will pop-up the LaTeX coding associated with them.

 

[fredrick08]

ya i see, thankyou very much, you have been a great help = ) much appreciated