Concepts about infinite potential well

  • Thread starter VHAHAHA
  • Start date
  • #1
58
0
1381506191751.jpg

As we know, the 1d infinite potential well has a stationary state. The function that depends on x onky is a sin function.
However, I don't understand the concept in this question. I have the answer of this question and this is not a homework. I am not asking for the answer so please don't put this post to the homework section.
I don't understand the part 2 of this question because the potential well should be in stationary state. Why i need to prove it in part 2?
Also, in part 3, i don't understand why there is a constant function. The fuction should be a product of sin function and exponential function. How come there is a constant function?
Thank you for your help
 

Answers and Replies

  • #2
1,948
200
The authors are asking you to expand a constant function in a Fourier sine series. It's a constant in the sense that it does not depend on the position for a fixed time t=0. For subsequent times, it won't be a constant anymore.
 
  • #3
58
0
The authors are asking you to expand a constant function in a Fourier sine series. It's a constant in the sense that it does not depend on the position for a fixed time t=0. For subsequent times, it won't be a constant anymore.

what is a fourier series:confused:
 
  • #4
jtbell
Mentor
15,755
3,965
what is a fourier series:confused:

A sum of sines and/or cosines whose frequencies are integer multiples of a fundamental frequency (if we're talking about a function of time) or whose wavenumbers are integer multiples of a fundamental. See for example

http://en.wikipedia.org/wiki/Fourier_series
 
  • #5
58
0
what is the relation between these two ?:(
 
  • #6
19
0
See, Simply put , stationary states of 1-d box are those in which the probability density (and also expectation values) are independent of time. do this : construct the time dependent wavefunction by stacking exponential time dependence alongside the sines. check out the probablity density. in this case multiplication of complex exponentail (in time) and its conjugate causes time dependence part to fizzle out, leaving only sine squared. ditto with avg. value of any operator.
QED
 
  • #7
jtbell
Mentor
15,755
3,965
what is the relation between these two ?:(

I'm guessing that by "these two" you mean Fourier series and the infinite square well.

The stationary states of the infinite square well (0 < x < a) are
$$\Psi_n(x,t) = \sqrt{\frac{2}{a}}\sin \left( \frac{n \pi x}{a} \right) e^{-i E_n t / \hbar}$$

Any state of the infinite square well can be written as a linear combination of these:
$$\Psi(x,t) = \displaystyle\sum_{n=1}^\infty {c_n \Psi_n(x,t)}$$

At t = 0:
$$\Psi(x,0) = \displaystyle\sum_{n=1}^\infty {c_n \sqrt{\frac{2}{a}}\sin \left( \frac{n \pi x}{a} \right)}$$

which is basically a Fourier series. For any function defined in the range 0 < x < a, you can find the coefficients ##c_n## which make this true. Put these coefficients into the linear combination for ##\Psi(x,t)## and it tells you how the wave function evolves in time from ##\Psi(x,0)##. Any decent QM textbook has the details.
 

Related Threads on Concepts about infinite potential well

  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
13
Views
2K
  • Last Post
Replies
1
Views
416
  • Last Post
Replies
5
Views
3K
  • Last Post
Replies
20
Views
10K
Replies
4
Views
1K
Replies
9
Views
1K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
10
Views
15K
Replies
9
Views
4K
Top