• Support PF! Buy your school textbooks, materials and every day products via PF Here!

Quantum mechanics hermite polynomials

  • Thread starter jc09
  • Start date
45
0
1. Homework Statement
Show that the one-dimensional Schr¨odinger equation
ˆ
(p^2/2m) ψ+ 1/2(mw^2)(x)ψ = En ψ
can be transformed into
(d^2/d ξ ^2)ψ+ (λn- ξ ^2) ψ= 0 where λn = 2n + 1.
using hermite polynomials


2. Homework Equations

know that dHn(X)/dX= 2nHn(x)

3. The Attempt at a Solution
I don't know how to start this question off at all
 
125
0
The question is a bit unclear to me. If I understand it correctly you should do the following:
1. get the first equation to the form
[tex]$ d^2\psi/dx^2 + 2m/\hbar^2(E-m\omega^2 x^2/2) \psi = 0$[/tex]
2. substitute [tex]$ \xi = x \sqrt{m\omega/\hbar}$[/tex]
3. you should get
[tex]$ d^2\psi/d\xi^2 + (2E/\hbar \omega - \xi^2) \psi = 0$[/tex]
4. now substitute [tex]$\psi = \varphi(\xi) \exp(-\xi^2/2)$[/tex] to get
5. [tex]$ d^2\varphi/d\xi^2 - 2\xi d\varphi/d\xi + (2E/\hbar\omega-1)\varphi = 0$[/tex]
This equation has solutions that diverge at infinity not faster than a polynomial - Hermite polynomials - only when [tex]$ 2E/\hbar\omega-1 = 2n$[/tex], where [tex]$ n = 0,1,2,...$[/tex]

Hope that helps!
 
45
0
Hey thats perfest thanks was stuck on how to get ξ into the equation.
 

Related Threads for: Quantum mechanics hermite polynomials

  • Posted
Replies
9
Views
4K
  • Posted
Replies
0
Views
2K
Replies
1
Views
1K
  • Posted
Replies
6
Views
1K
Replies
1
Views
2K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top