Finding potential to satisfy SE

  • Thread starter gfd43tg
  • Start date
  • #1
gfd43tg
Gold Member
953
49

Homework Statement


upload_2015-2-10_15-44-39.png


Homework Equations




The Attempt at a Solution



(a) Well, I just isolate A, so
$$A = \Psi (x,t) e^{a[(mx^{2}/ \hbar) + it]}$$

I am not sure if this is what is meant, seems too obvious.

(b) So I know the Schrödinger equation can be written
$$ i \hbar \frac {\partial \Psi}{ \partial t} = \Big [ - \frac {\hbar}{2m} \frac {\partial^{2}}{\partial x^{2}} + V \Big ] \Psi $$

So I take the given wave function,
$$\Psi = Ae^{-a[\frac {mx^{2}}{\hbar} + it]}$$

And find the derivatives with respect to x and t,

$$ \frac {\partial \Psi}{\partial t} = -A[a(\frac {mx^{2}}{\hbar}) + i]e^{-a[\frac {mx^{2}}{\hbar} + it]} $$
$$ \frac {\partial \Psi}{\partial x} = -A[a(\frac {2mx}{\hbar}) + it]e^{-a[\frac {mx^{2}}{\hbar} + it]} $$
$$ \frac {\partial^{2} \Psi}{\partial x^{2}} = A[a^{2}(\frac {4m^{2}x^{2}}{\hbar^{2}}) + i^{2}t^{2}]e^{-a[\frac {mx^{2}}{\hbar} + it]} $$

And I substitute back into the SE,
$$ i \hbar(-A[a(\frac {mx^{2}}{\hbar}) + i]e^{-a[\frac {mx^{2}}{\hbar} + it]}) = - \frac {\hbar^{2}}{2m} \Big( A[a^{2}(\frac {4m^{2}x^{2}}{\hbar^{2}}) + i^{2}t^{2}]e^{-a[\frac {mx^{2}}{\hbar} + it]} \Big ) + V( Ae^{-a[\frac {mx^{2}}{\hbar} + it]}) $$

From here I can isolate V
 
Last edited:

Answers and Replies

  • #2
Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
16,829
6,652
(a) No, this is not what is intended. You do not know the wave function up to the constant A so you obviously cannot have an expression for A involving the wave function. You need to apply some criteria that the wave function should fulfil. What such criteria do you know?

(b) Your post seems incomplete as it ends "So I". What did you intend to say?
 
  • #3
DEvens
Education Advisor
Gold Member
1,203
457
Overall: Check your derivatives.

Re: Finding A. Remember what the wave function means. When you take psi* times psi, you get the probability of finding the particle in the range dx. So, what is the total probability of finding the particle somewhere? And so, how can you determine A?

For the rest: Can you see any factors to divide out of your last equation?
 
  • #4
Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
16,829
6,652
Also, the derivatives are not done correctly. Recheck what you have in the argument of the exponential.
 

Related Threads on Finding potential to satisfy SE

Replies
2
Views
550
Replies
1
Views
650
Replies
8
Views
8K
Replies
2
Views
1K
Replies
7
Views
858
Top