Difference between revisions of "Quantum Qual Problems"

From New IAC Wiki
Jump to navigation Jump to search
Line 16: Line 16:
 
In our case, using separation of variables, we will get 3 differential equations for x, y and z. W(x,y,z)=w(x)w(y)w(z)<br>
 
In our case, using separation of variables, we will get 3 differential equations for x, y and z. W(x,y,z)=w(x)w(y)w(z)<br>
  
<math>\frac{d^2 w(x)}{dx^2} - c^2 w(x) = 0</math>(1)<br>
+
<math>\frac{d^2 w(x)}{dx^2} - m^2 w(x) = 0</math>(1)<br>
 
The same will be for y and z.<br>
 
The same will be for y and z.<br>
  
 
Solution of equation (1) is following <br>
 
Solution of equation (1) is following <br>
<math>w(x) = A\sin(cx)+B\cos(cx)</math><br>
+
<math>w(x) = A\sin(mx)+B\cos(mx)</math><br>
<math>w(y) = C\sin(cy)+D\cos(cy)</math><br>
+
<math>w(y) = C\sin(ky)+D\cos(ky)</math><br>
<math>w(z) = E\sin(cz)+F\cos(cz)</math><br>
+
<math>w(z) = E\sin(qz)+F\cos(qz)</math><br>
  
 
* Applying B. C. at x=y=z=0 wave function should be zero, that means B=D=F=0. We have<br>
 
* Applying B. C. at x=y=z=0 wave function should be zero, that means B=D=F=0. We have<br>
  
<math>w(x) = A\sin(cx) </math><br>
+
<math>w(x) = A\sin(mx) </math><br>
<math>w(y) = C\sin(cy) </math><br>
+
<math>w(y) = C\sin(ky) </math><br>
<math>w(z) = E\sin(cz) </math><br>
+
<math>w(z) = E\sin(qz) </math><br>
  
Also, w(a)=0 which gives <math>A\sin(cx)=0, cx=\pi n</math>
+
Also, w(a)=0 which gives <math>A\sin(ma)=0, m=\frac{\pi n}{a}</math>,  <math>C\sin(kb)=0, k=\frac{\pi n}{b}</math>,  <math>E\sin(qc)=0, q=\frac{\pi n}{c}</math>

Revision as of 03:01, 16 August 2007

1.) Given a quantum mechanical particle of mass [math]M[/math] confined inside a box of sides [math]a,b,c[/math]. The particle is allowed to move freely between [math]0 \lt x \lt a, 0\lt y\lt b [/math] and [math]0\lt z\lt c[/math].

  • Use the time-independent Schrodinger equation for this problem to obtain the general form for the eigenfunctions of the particle
  • Now apply boundary conditions to obtain the specific eigenfunctions and eigenenergies for this specific problem.
  • Assume [math]a=b=c[/math] and find the first 6 eigenenergies of the problem in terms of the box side length ([math]a[/math]), the particle mass ([math]M[/math]) and standard constants. What are their quantum number? Make a sketch of the eigenvalue spectrum, a table listing these eigenenergies and the quantum numbers of all the states that correspond to them.


Solution:


2.)

  • [math] [- \frac{h^2}{2m}\Delta^2 + V]W(x,y,z)=E W(x,y,z) [/math]

In our case, using separation of variables, we will get 3 differential equations for x, y and z. W(x,y,z)=w(x)w(y)w(z)

[math]\frac{d^2 w(x)}{dx^2} - m^2 w(x) = 0[/math](1)
The same will be for y and z.

Solution of equation (1) is following
[math]w(x) = A\sin(mx)+B\cos(mx)[/math]
[math]w(y) = C\sin(ky)+D\cos(ky)[/math]
[math]w(z) = E\sin(qz)+F\cos(qz)[/math]

  • Applying B. C. at x=y=z=0 wave function should be zero, that means B=D=F=0. We have

[math]w(x) = A\sin(mx) [/math]
[math]w(y) = C\sin(ky) [/math]
[math]w(z) = E\sin(qz) [/math]

Also, w(a)=0 which gives [math]A\sin(ma)=0, m=\frac{\pi n}{a}[/math], [math]C\sin(kb)=0, k=\frac{\pi n}{b}[/math], [math]E\sin(qc)=0, q=\frac{\pi n}{c}[/math]