Solving for Inner Products in an Infinitely Deep q Well

[rshalloo]

Homework Statement

See attachment

Homework Equations

for an infinitely deep q well of width L
$\psi(x)=\sqrt{\frac{2}{L}}Sin\frac{4\pi x}{L}$
$<\psi_{n}|\psi_{m}>=0$
$<\psi_{n}|\psi_{n}>=1$

The Attempt at a Solution

Taking the inner product;

$<\psi|\psi>=a<\psi|\psi_{1}>+b<\psi|\psi_{2}>$

$<\psi|\psi>=a<\psi_{1}|\psi_{1}>+b<\psi_{2}|\psi_{2}>$

$1=a+b$

from the second part of the question 3a=b and thus a=.25 b=.75

I kind of went out on a limb with this and I'm not sure if its correct. It seems very little work for the available marks and I'm just not sure if the second part is ok?

 

[mark726]

Thank you for your post. Your approach to solving this problem is correct, but there are a few things that could be improved upon.

Firstly, when taking the inner product, you should use the complex conjugate of one of the wavefunctions. In this case, it would be <\psi|\psi_{1}^*>+<\psi|\psi_{2}^*>. This is because the inner product requires one of the wavefunctions to be in the bra ( <\psi| ) and the other in the ket ( |\psi_{n}> ) notation.

Secondly, your solution for a and b is incorrect. The correct solution is a=0.25 and b=0.75, as you stated in the second part of your post. This is because the inner product of two orthogonal wavefunctions is always equal to zero, and in this case, <\psi_{1}|\psi_{2}>=0.

Lastly, I would suggest showing more steps in your solution to demonstrate your understanding of the problem. For example, you could show how you arrived at the equation a+b=1, and then substitute in the given values for a and b to show how you arrived at a=0.25 and b=0.75.

Overall, your solution is correct, but I would recommend revising and showing more steps to fully demonstrate your understanding of the problem. Keep up the good work!