Why does the wave function have to be normalizabled

In summary, the wave function must be normalizable and go to zero at infinity in order to accurately describe the position of a particle. This is because the probability density must be integrated over all space and equal to one, meaning the particle must be somewhere. If the wave function does not go to zero at infinity, the probability of the particle's position becomes infinite. Normalizing the wave function ensures that the maximum possible probability is 1.
  • #1
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why does the wave function have to be normalizable, and why does it have to go to 0 and x approaches positive/negative infinity and y approaches positive/negative infinity ?
 
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  • #2
Think about the physics involved. Here's a simple and intuitive explanation. The particle described by the wave function has to be found somewhere so the wave function must be normalizable. From this it follows that it must go to zero as x->inf.
 
  • #3
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i still don't get it...
 
  • #4
Could you be more precise about what you don't get?

[tex]\psi \psi*[/tex] is the probability density. Since the particle is certainly somewhere the probability density integrated over all space must give 1. Or in other words the particle has to be somewhere. If this is to hold psi has to go to zero at infinity.
 
  • #5
when you normalize the wave function to be 1, do you mean that you do it to make sure that the biggest limit is 1? and what does 0 have to do with infinity? i only see in my textbook that the wave function has no meaning, but its square means probability, so i think I'm confused on what does the boundary limits on the integral mean...

sorry about that, I'm slow at learning things... :P
 
  • #6
Ok you normalise the integral of psi^2 between the infinities (+ve and -ve)to be equal to one, i.e. you normalize so that the probability of the particle being somewhere between the origin and the infinities of all spatial dimensions is equal to one. Sensible no? Ok, if the wave function does not go to zero at the infinities then the probability of the particle being somwhere in space is equal to infinity. The reason for that is that if you integrate the probabilty density function over all space from infinity to infinty you get infinity, if the function does not go to zero at the infinities..., hehe, and the maximum possible value for a probabilty is 1... so the wavefuntion has to disappear at infinity.
 
  • #7
i see now~ thank you very much for clearing that up! :)
 

1. Why is it important for the wave function to be normalizable?

The wave function represents the probability amplitude of a particle in quantum mechanics. For this probability to make physical sense, it must be finite and have a total probability of 1. Normalization ensures that the probability of finding the particle anywhere in space is equal to 1, making the wave function physically meaningful.

2. What happens if the wave function is not normalized?

If the wave function is not normalized, it means that the total probability of finding the particle in any location is not 1. This would violate the fundamental principle of quantum mechanics, which states that the total probability of all possible outcomes must equal 1. In other words, the wave function would not accurately describe the behavior of the particle.

3. How is normalization of the wave function achieved?

Normalization of the wave function is achieved by dividing the wave function by its norm, which is the square root of the integral of its absolute square. This ensures that the total probability of finding the particle anywhere in space is 1.

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

No, the wave function must always be normalized to a value of 1. This is a fundamental principle of quantum mechanics and is necessary for the wave function to accurately represent the probability of finding a particle in a given location.

5. What are the consequences of not normalizing the wave function?

If the wave function is not normalized, it can lead to incorrect predictions and interpretations in quantum mechanics. For example, it can result in probabilities exceeding 1 or negative probabilities, which do not have physical meaning. Moreover, normalization is necessary for the wave function to be used in mathematical calculations and to make accurate predictions about the behavior of particles.

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