Normalization of wave function

In summary, the conversation discusses a wave function and a time-independent potential well in one dimension. The question at hand is to find the probability that a proton is located between two specific points. The individual discussing the problem is unsure about the bounds for normalizing the wave function, but it is clarified that integrating from -L/2 to +L/2 is sufficient. They also confirm that to find the probability, integration can be done over the given bounds.
  • #1
Safder Aree
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1

Homework Statement


I have the wave function Ae^(ikx)*cos(pix/L) defined at -L/2 <= x <= L/2. and 0 for all other x.

The question is:
A proton is in a time-independent one-dimensional potential well.What is the probability that the proton is located between x = − L/4 and x = L/4 ?

Homework Equations


∫ψψ* = 1

The Attempt at a Solution


I know i have to normalize this first. Should I be normalizing with the bounds being -L/2 and L/2 or should the bounds be -L/4 and L/4.
A^2∫cos^2(πx/L)dx =1
(1/2)A^2[x+ L/2π (sin(2πx/L)] evaluated at some bounds.

I actually evaluated it both ways one answer gives me A=√2/L at L/2,-L/2 and the other one gives me A = √2
π/L which makes more sense (at -L/4,L/4)

But i just want to make sure i am approaching this right way.
Thank you.
 
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  • #2
Actually, the normalization in one dimension is formally over all space, i.e. from -∞ to +∞, but since the wavefunction is zero outside the box, you would be adding a whole bunch for zeroes if you went outside it. Therefore it suffices to integrate from -L/2 to +L/2, i.e. over all space where there is non-zero probability.
 
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  • #3
I actually re did it and got A=√2/L. Thank you for the clarification. Now to find the probability I can just integrate over the other bounds right?
 
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  • #4
Right. Make sure the number you get is less than 1.
 
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1. What is the normalization of a wave function?

The normalization of a wave function is the process of adjusting its amplitude or overall magnitude so that the total probability of finding the particle in all possible states is equal to 1.

2. Why is normalization important in quantum mechanics?

Normalization is important in quantum mechanics because it ensures that the probability of finding a particle in any possible state is well-defined and consistent with the principles of probability theory. It also ensures that the wave function represents a physically observable quantity.

3. How do you normalize a wave function?

To normalize a wave function, you divide the original wave function by the square root of the integral of its absolute square over all space. This process ensures that the total probability of finding the particle is equal to 1.

4. Can a wave function be normalized to a value other than 1?

No, the normalization constant for a wave function must always be chosen so that the total probability is equal to 1. This is a fundamental principle in quantum mechanics and ensures that the wave function represents a physically meaningful quantity.

5. What happens if a wave function is not normalized?

If a wave function is not normalized, it does not represent a physically observable quantity. This can lead to incorrect predictions and violate the principles of probability theory. Normalization is an essential step in the analysis of quantum mechanical systems.

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