Peak of psi squared and <x> not same?

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    Peak Psi
In summary: I observe a large number of particles prepared in the same way, while <x_max> is the x such that dP/dx = 0.
  • #1
mmwave
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While working on an infinite barrier problem I was surprised to find that the location of |psi|^2 = 0.30 a while <x> is 0.32 a, where a is the width of the well.

It's often said that |psi|^2 tells one about the probabilty of finding a particle in a location x. Yet the expectation of x, <x> is a slightly different value. How do I reconcile the difference between these two values?

If <x> is the average of values if I observe a large number of particles prepared in the same way, then what exactly is the peak of |psi|^2 ?
 
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  • #2
Originally posted by mmwave
It's often said that |psi|^2 tells one about the probabilty of finding a particle in a location x. Yet the expectation of x, <x> is a slightly different value. How do I reconcile the difference between these two values?

This isn't anything mysterious about quantum mechanics; it's well known in statistics that the most probable value of a random variable is generally not the the same as the mean (or expected) value of the variable. In fact it rarely is, Gaussian distributions being an exception. I'm not sure what needs "reconciling" here...?
 
  • #3
That's it, just a heuristic explanation for why the most probable does not equal the mean. Since nearly all the distributions I have ever dealt with are treated as gaussian, I forget those little details like the mean is not always equal to the most probable value.

I imagine that as the distribution gets broader or more assymmetrical the mean can be further from the most probable value?
 
  • #4
Originally posted by mmwave
I imagine that as the distribution gets broader or more assymmetrical the mean can be further from the most probable value?

Yes... imagine, for instance, something that is mostly Gaussian, except with a thin spike far off to the side, that's just slightly higher than the main Gaussian. The most probable value will be the spike, but as far as the mean is concerned, the spike makes almost no difference: the mean will still be at the center of the Gaussian.

Of course, you can cook up all kinds of screwy functions, and in general, the mean and the maximum will have nothing to do with each other. After all,

x_max is the x such that dP/dx = 0
<x> = integral^b_a x P(x) dx / (b-a)

The two aren't intrinsically related.
 

1. What is the "peak of psi squared" and how is it related to ?

The "peak of psi squared" refers to the maximum value of the wave function squared, which is a measure of the probability of finding a particle at a specific position. represents the average position of the particle, and the two are related through the equation = ∫x|ψ|^2dx, where ψ is the wave function.

2. Why is the peak of psi squared not always the same as ?

The peak of psi squared and are not always the same because the wave function can have different shapes and distributions, even for particles in the same system. This means that the maximum probability of finding a particle at a certain position (peak of psi squared) can occur at a different position than the average position of the particle ().

3. Can the peak of psi squared be outside of the range of ?

Yes, it is possible for the peak of psi squared to be outside of the range of . This can occur when the wave function has multiple peaks or is asymmetric, causing the maximum probability of finding the particle to be at a position that is not the average position.

4. How does the uncertainty principle relate to the peak of psi squared and ?

The uncertainty principle states that it is impossible to know the exact position and momentum of a particle simultaneously. This means that, while the peak of psi squared represents the most probable position of the particle, there is always some uncertainty in its exact location. This uncertainty is reflected in the spread of the wave function, which can affect the value of .

5. Are there any real-world applications for understanding the relationship between the peak of psi squared and ?

Yes, understanding the relationship between the peak of psi squared and is important in many scientific fields, such as quantum mechanics, chemistry, and material science. It helps us to understand the behavior of particles and their interactions, and can also be applied in technologies such as quantum computing and nanotechnology.

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