Constructing Wave Function for Particle in 1D Box: Is it Unique?

[eit32]

For a particle in a 1-dimensional box confined by 0<x<a.
a)Construct a wave function phi(x)=psi(x,t=0) such that when an energy measurement is made on the particle in this state at t=0, the following energy values are obtained with the probabilities shown:

Energy E_n Obtained : E_1 E_3 E_5
Probability of Obtaining E_n : 0.5 0.4 0.1

b) Is this answer unique? Why or why not? Illustrate with an example.

 

[CompuChip]

For a), calculate the eigenfunctions \psi_n. What does a general wave function look like? What is the probability of finding E_n when measuring the energy (this will also answer b))

 

[eit32]

I don't know how to go about finding the eigenfunctions, that's part of the problem

 

[CompuChip]

OK, so that needs some explanation. But let's focus on the important part of the question first: suppose you have the eigenfunctions \psi_n, such that
\hat H \psi_n(x) = E_n \psi_n(x)
Then can you answer the rest of my questions? (just express the answers in terms of \psi_n)

 

[eit32]

a general wave function is usually in the form of either an exponential or sines and cosines, and I'm not really sure about the probability

 

[merryjman]

If I remember correctly, probability functions are simply the square of the wave functions, normalized so that the integral of the prob. func. over all space is 1.

As far as the wave function itself, maybe start with this: since you are aware that these are usually sines/cosines, what sort of sine/cosine will be zero at x=0 and x=a, since the function can't exist at the boundary?

 

[CompuChip]

If you use Griffith's book, you'll find in equation [2.16] the general (time-independent) solution
\Psi(x) = \sum_n c_n \psi_n(x).
Also he explains at the bottom of page 36 (in my 2nd edition) that
"As we'll see in chapter 3, what |c_n|^2 tells you is the probability that a measurement of the energy would yield the value E_n."

(If you don't have this book, go buy it; IMO it is by far the best introductory QM out there!)
Now can you see how to apply this to the problem at hand?

 

[droedujay]

General 1-D Time independent Schrodingers equation

sqrt(2/a)*sin(n*pi*x/a)