Quantum mechanics probability

In summary: First indication to use |\psi(x,t)|^2 is that this quantity is normalised such that the integral is = 1 which means that the probablity to find the particle anywhere is 100%.
  • #1
matematikuvol
192
0
In the one moment you accept something in the other you're not sure why? Why probability density in quantum mechanics is defined as ##|\psi|^2## and no just ##|\psi|## if we know that ##|\psi|## is also positive quantity.
 
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  • #2
matematikuvol said:
In the one moment you accept something in the other you're not sure why? Why probability density in quantum mechanics is defined as ##|\psi|^2## and no just ##|\psi|## if we know that ##|\psi|## is also positive quantity.
Because - in general, ##\psi## is a complex number ... go look at how complex numbers work. i.e., ##|\psi(\vec{r},t)|## is not, in general, positive everywhere.
Consider a particle in a 1D unit box, prepared in the n=2 state.

Mainly the model is the way it is because that is what nature does.
Just taking the modulus does not give the correct statistics.
 
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  • #3
Of course [itex]|\psi(t,\vec{x})|[/itex] is a positive semidefinite real expression everywhere, since the modulus of a complex number is a positive real number (or 0 if the number itself is 0).

Born's rule is, as far as we know, one of the fundamental postulates of quantum theory. It has not yet been deduced from other more fundamental postulates yet. So we cannot explain, why this interpretation of the wave function works to describe nature.

Recently, however it has been tested experimentally that it holds with very high accuracy:

http://physicsworld.com/cws/article/news/2010/jul/22/quantum-theory-survives-its-latest-ordeal
 
  • #4
I believed that theory is consistent. But why Bohm took ##|\psi(x,t)|^2##? Is there some bond with
[tex]\vec{j}=\frac{\hbar}{2im}(\psi^*\frac{d}{dx}\psi-\psi\frac{d}{dx}\psi^*)[/tex]
 
  • #5
A Google search for "Born rule" led me to the Wikipedia page

http://en.wikipedia.org/wiki/Born_rule

which refers to Born's 1954 Nobel Prize lecture

http://nobelprize.org/physics/laureates/1954/born-lecture.pdf

in which he talks about the origin of his famous rule.

Max Born said:
Again an idea of Einstein’s gave me the lead. He had tried to make the duality of particles - light quanta or photons - and waves comprehensible by interpreting the square of the optical wave amplitudes as probability density for the occurrence of photons. This concept could at once be carried over to the psi-function: |psi|2 ought to represent the probability density for electrons (or other particles)

Classically, the square of the electric field (or of the magnetic field) is proportional to the energy density carried by the field, and therefore also to the number-density of photons (photons per m3)in the field. For a weak field (small number of photons), we have to talk about probability rather than number-density.
 
  • #6
First indication to use [itex] |\psi(x,t)|^2[/itex] is that this quantity is normalised such that the integral is = 1 which means that the probablity to find the particle anywhere is 100%.

The second indication is that the density

[tex]\rho = |\psi|^2[/tex]

plus the current

[tex]\vec{j}=\frac{\hbar}{2im}(\psi^*\nabla\psi-\psi\nabla\psi^*)[/tex]

fulfill a continuity equation

[tex]\frac{\partial}{\partial t}\rho + \nabla \vec{j} = 0[/tex]

which means that the probablity is conserved. This follows in a rather general way using the Lagrangian corresponding to the Schrödinger equation plus Noether's theorem. It is not special for Bohm's interpretation.

All this works for [itex] |\psi(x,t)|^2[/itex] not for [itex] |\psi(x,t)|[/itex]
 
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  • #7
tom.stoer said:
First indication to use [itex] |\psi(x,t)|^2[/itex] is that this quantity is normalised such that the integral is = 1 which means that the probablity to find the particle anywhere is 100%.

Well I agree with second reason. But you can normalized state in such way that
[tex]\int^{\infty}_{-\infty}|\psi(x,t)|dx=1[/tex]
Right?
And the second one. Why current is defined like
[tex]\vec{j}=\frac{\hbar}{2im}(\psi^*\frac{d}{dx}\psi-\psi\frac{d}{dx}\psi^*)[/tex]
 
  • #8
matematikuvol said:
Well I agree with second reason. But you can normalized state in such way that
[tex]\int^{\infty}_{-\infty}|\psi(x,t)|dx=1[/tex]
Right?
No, because we are using the space L^2(R^3) for the space of wavefunctions. Not every L^2 function is also a L^1 function. We can't use L^1, because it's not a Hilbert space.

And the second one. Why current is defined like
[tex]\vec{j}=\frac{\hbar}{2im}(\psi^*\frac{d}{dx}\psi-\psi\frac{d}{dx}\psi^*)[/tex]
Because this definition turns out to give you a conserved current.
 
  • #9
rubi said:
No, because we are using the space L^2(R^3) for the space of wavefunctions. Not every L^2 function is also a L^1 function. We can't use L^1, because it's not a Hilbert space.
Correct; L^2 is the only Lp-Banach-Space which is also a Hilbert space.

rubi said:
Because this definition turns out to give you a conserved current.
Using Noether's theorem this current can be derived.

matematikuvol said:
But you can normalized state in such way that
[tex]\int^{\infty}_{-\infty}|\psi(x,t)|dx=1[/tex]
Right?
No.

You can normalize the density for one time t, but time evolution (Schrödinger equation) will not respect this normalization, i.e. it will be violated for t' > t.

As I said
tom.stoer said:
all this works for [itex] |\psi(x,t)|^2[/itex] not for [itex] |\psi(x,t)|[/itex]
 
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  • #10
tom.stoer said:
Using Noether's theorem this current can be derived

Yeah, given Schrodinger's equation alone, with no assumption about the meaning of [itex]\vert \Psi \vert^2[/itex], you can prove that the latter is a conserved quantity (when integrated over all space). So it's pretty natural to interpret it as something conserved: probability, or charge, or energy...
 
  • #11
stevendaryl said:
Yeah, given Schrodinger's equation alone, with no assumption about the meaning of [itex]\vert \Psi \vert^2[/itex], you can prove that the latter is a conserved quantity ...
Yes, you can prove that, but you cannot derive it w/o Noether's theorem; you have to guess it.
 
  • #12
Hello, I have a question concerning normalization. I start from integral |psi|^2 dV = 1, which means somehow that the particle should be in the universe with probability 1. Now using the the Gauss theorem in such a way that if we have a vector function

Fi: div Fi = |psi|^2,
integral |psi|^2 dV = integral div Fi dv =(Gauss) integral Fi dA = 1 so the normalization is transformed from volumetric to surface integral.

The unit of psi is [psi]=1/m^(3/2) so [psi^2]=1/m^3 and vector [Fi] = 1/m^2.

So we prepared a "surface probability density vector" from the scalar volumetric probability density |psi|^2.

I would like to know whether this trick with mathematics could have some physical meaning at all? It looks like that the internal probability appeared on the surface of the universe somehow, also I am aware that probability vector sounds quite unphysical.

Thank you for your answer.
 

1. What is the role of probability in quantum mechanics?

In quantum mechanics, probability plays a fundamental role in understanding the behavior of particles at the atomic and subatomic level. This is because the behavior of particles in quantum mechanics is described by wave functions, which give the probability of finding a particle at a given location or with a given momentum. The uncertainty principle also states that it is impossible to know both the position and momentum of a particle simultaneously, further emphasizing the role of probability in quantum mechanics.

2. How is probability calculated in quantum mechanics?

In quantum mechanics, probability is calculated using the wave function, which is described by Schrodinger's equation. The square of the wave function, known as the probability density, gives the probability of finding a particle at a specific location. The probability of an event occurring is then calculated by integrating the probability density over a particular region.

3. What is the difference between classical and quantum probability?

The main difference between classical and quantum probability is that classical probability deals with events that have definite outcomes, while quantum probability deals with events that have a range of possible outcomes. In classical probability, the probability of an event is always a specific value between 0 and 1, while in quantum probability, the probability is described by a wave function that can take on any value between 0 and 1.

4. How is probability related to the measurement problem in quantum mechanics?

In quantum mechanics, the measurement problem refers to the issue of how a particle's wave function collapses into a definite state when it is observed. Probability is related to this problem because it is the only way to predict the outcome of an observation. The wave function gives the probability of a particle being in a particular state, and when the particle is observed, the wave function collapses to that state with a certain probability.

5. Can probability in quantum mechanics be visualized?

No, probability in quantum mechanics cannot be visualized in the same way that we can visualize probabilities in classical mechanics. This is because the wave function in quantum mechanics is a complex mathematical concept that does not have a direct visual representation. However, the probability density can be visualized as a graph, with the height of the graph representing the probability of finding a particle at a specific location.

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