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

In summary: So the eigenvalues are E_n = (sqrt(2/a)*sin(n*pi*x/a))/(h(x))and the probabilities are P_n(E_n) = (1-sqrt(2/a)*sin(n*pi*x/a))/(h(x))So the wave function is unique, and it has the following formPhi(x) = (sqrt(2/a)*sin(n*pi*x/a))/(h(x))
  • #1
eit32
21
0
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.
 
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  • #2
For a), calculate the eigenfunctions [itex]\psi_n[/itex]. What does a general wave function look like? What is the probability of finding [itex]E_n[/itex] when measuring the energy (this will also answer b))
 
  • #3
I don't know how to go about finding the eigenfunctions, that's part of the problem
 
  • #4
OK, so that needs some explanation. But let's focus on the important part of the question first: suppose you have the eigenfunctions [itex]\psi_n[/itex], such that
[tex]\hat H \psi_n(x) = E_n \psi_n(x)[/tex]
Then can you answer the rest of my questions? (just express the answers in terms of [itex]\psi_n[/itex])
 
  • #5
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
 
  • #6
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?
 
  • #7
If you use Griffith's book, you'll find in equation [2.16] the general (time-independent) solution
[tex]\Psi(x) = \sum_n c_n \psi_n(x)[/tex].
Also he explains at the bottom of page 36 (in my 2nd edition) that
"As we'll see in chapter 3, what [itex]|c_n|^2[/itex] tells you is the probability that a measurement of the energy would yield the value [itex]E_n[/itex]."

(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?
 
  • #8
General 1-D Time independent Schrodingers equation

sqrt(2/a)*sin(n*pi*x/a)
 

1. What is a particle in a 1D box?

A particle in a 1D box is a theoretical model used in quantum mechanics to study the behavior of a particle confined within a one-dimensional space. The box represents the boundaries that the particle cannot pass through, and the particle's wave function describes its probability of being found at a particular location within the box.

2. Why is the wave function for a particle in a 1D box important?

The wave function for a particle in a 1D box is important because it allows us to understand the behavior of quantum particles in a confined space. It also helps us to calculate various properties of the particle, such as its energy levels and probability of being found at a certain location within the box.

3. Is the wave function for a particle in a 1D box unique?

Yes, the wave function for a particle in a 1D box is unique. This means that for a given set of boundary conditions, there is only one wave function that satisfies the Schrödinger equation and describes the particle's behavior within the box.

4. What are the boundary conditions for a particle in a 1D box?

The boundary conditions for a particle in a 1D box are that the wave function must go to zero at the boundaries of the box. This means that the particle cannot exist outside of the box, and it also ensures that the wave function is continuous.

5. How is the wave function for a particle in a 1D box constructed?

The wave function for a particle in a 1D box is constructed by solving the Schrödinger equation for the given boundary conditions. This involves using mathematical techniques such as separation of variables and applying the boundary conditions to determine the specific form of the wave function. The resulting wave function is then normalized to ensure that the total probability of finding the particle within the box is equal to 1.

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