- #1
quellcrist
- 5
- 0
1.
##<x>= \int_{0}^{a}x\left | \psi \right |^{2}dx##
##\psi (x)=\sqrt{\frac{2}{a}}\sin\frac{n\pi x}{a}##
then ##<x>= \frac{2}{a} \int_{0}^{a}x \sin\frac{n\pi x}{a}dx##
2. Homework Equations
1) ##y=\frac{n\pi x}{a}## then ##dy=\frac{n\pi}{a}dx##
and
2)
##y=\frac{n\pi x}{a}## then ##dx=\frac{a}{n\pi}dy##
then
##\psi (x)=\sqrt{\frac{2}{a}} \sin(y)##
##<x>= \frac{2}{a}\int_{0}^{a=n\pi}y \sin^{2}ydy \times \frac{a}{n\pi} \times \frac{a}{n\pi}##
3. The Attempt at a Solution
I don't need help solving the general problem for the expectation value of x...I have the solution manual. The question I have is about how/why they chose to solve the integral this way by substituting y for (n*pi*x)/(a)? I understand how 1) works but I need help clarifying how 2) works.
I need a general walkthrough of why they are doing this integral this way.
Thank you
<Moderator's note: formatting tidied up. OP, please make sure your posts are readable and use the proper LaTeX tags>
##<x>= \int_{0}^{a}x\left | \psi \right |^{2}dx##
##\psi (x)=\sqrt{\frac{2}{a}}\sin\frac{n\pi x}{a}##
then ##<x>= \frac{2}{a} \int_{0}^{a}x \sin\frac{n\pi x}{a}dx##
2. Homework Equations
1) ##y=\frac{n\pi x}{a}## then ##dy=\frac{n\pi}{a}dx##
and
2)
##y=\frac{n\pi x}{a}## then ##dx=\frac{a}{n\pi}dy##
then
##\psi (x)=\sqrt{\frac{2}{a}} \sin(y)##
##<x>= \frac{2}{a}\int_{0}^{a=n\pi}y \sin^{2}ydy \times \frac{a}{n\pi} \times \frac{a}{n\pi}##
3. The Attempt at a Solution
I don't need help solving the general problem for the expectation value of x...I have the solution manual. The question I have is about how/why they chose to solve the integral this way by substituting y for (n*pi*x)/(a)? I understand how 1) works but I need help clarifying how 2) works.
I need a general walkthrough of why they are doing this integral this way.
Thank you
<Moderator's note: formatting tidied up. OP, please make sure your posts are readable and use the proper LaTeX tags>