# Does expansion a(t) affect QM probability density?

• Jimster41
In summary: In quantum mechanics, the wavefunction is commutative, so the product of two wavefunctions is the same as the product of their individual wavefunctions. But in classical mechanics, the wavefunction is not commutative, so the product of two wavefunctions may not be the same as the product of their individual wavefunctions.
Jimster41
I'm picturing a comoving particle, meaning at rest with respect to CMB? Does expansion affect the probability density, density current? I can't see how it wouldn't? But then it seems like there would have to be a uniform negative probability density current everywhere.

Last edited:
I guess the definition of probability density current should be modified then. I'm not aware of setups where this would actually be relevant.

This is the equation I am trying to understand the meaning of.

$j\quad =\quad \frac { \hbar }{ 2mi } \left( { \Psi }^{ * }\frac { \partial \Psi }{ \partial x } -\Psi \frac { \partial { \Psi }^{ * } }{ \partial x } \right)$

The part inside the parenthesis is what i would like to get a feel for. I know it describes the difference between the product of of the wave function and the partial derivative of it's conjugate, and the product of the conjugate and the partial of the wavefunction (the "reversed product").

The difference would represent... the spatial non-commutativity of the wavefunction?

Not that it's important but this is the paper I am trying to read understand some of.

http://arxiv.org/pdf/quant-ph/0105112v1.pdf
On Koopman-von Neumann Waves
D. Mauro
(Submitted on 23 May 2001)
In this paper we study the classical Hilbert space introduced by Koopman and von Neumann in their operatorial formulation of classical mechanics. In particular we show that the states of this Hilbert space do not spread, differently than what happens in quantum mechanics. The role of the phases associated to these classical "wave functions" is analyzed in details. In this framework we also perform the analog of the two-slit experiment and compare it with the quantum case.

The expansion causes particle production. I think the rate is low, bu it is similar to why the black hole produces Hawking radiation, in the sense that neither the expanding universe not the interior of the black hole is stationary.

http://arxiv.org/abs/1205.5616
Particle creation and particle number in an expanding universe
Leonard Parker

Jimster41
Jimster41 said:
This is the equation I am trying to understand the meaning of.

$j\quad =\quad \frac { \hbar }{ 2mi } \left( { \Psi }^{ * }\frac { \partial \Psi }{ \partial x } -\Psi \frac { \partial { \Psi }^{ * } }{ \partial x } \right)$

The part inside the parenthesis is what i would like to get a feel for. I know it describes the difference between the product of of the wave function and the partial derivative of it's conjugate, and the product of the conjugate and the partial of the wavefunction (the "reversed product").

Intuitively, this formula is basically the particle velocity times the probability density. Recall that the momentum operator is ##\frac{\hbar}{i} \frac{\partial}{\partial x}##. We divide by ##m## to get the velocity.

Jimster41 said:
The difference would represent... the spatial non-commutativity of the wavefunction?

The difference is just needed to make the overall expression real-valued.

Jimster41

## 1. How does the expansion of the universe affect the quantum mechanical probability density?

The expansion of the universe does not directly affect the quantum mechanical probability density. The probability density is determined by the wave function, which is independent of the scale of the universe. However, the expansion of the universe can indirectly impact the probability density through the redshifting of light and the changing of distances between objects.

## 2. Does the expansion of the universe change the probability of quantum events occurring?

No, the expansion of the universe does not change the probability of quantum events occurring. The probability of a quantum event is determined by the wave function, which is not affected by the expansion of the universe.

## 3. Can the expansion of the universe affect the uncertainty principle in quantum mechanics?

No, the expansion of the universe does not affect the uncertainty principle in quantum mechanics. The uncertainty principle is a fundamental principle of quantum mechanics and is not dependent on the scale of the universe.

## 4. Are there any observable effects of the expansion of the universe on quantum mechanical systems?

Yes, there are observable effects of the expansion of the universe on quantum mechanical systems. As the universe expands, the wavelengths of light from distant objects are stretched, causing a redshift. This can affect the measurable properties of these objects, such as their energy levels, which can in turn impact the probability density of quantum events involving these objects.

## 5. How does the expansion of the universe impact the concept of quantum entanglement?

The expansion of the universe does not directly impact the concept of quantum entanglement. However, it can make it more difficult to observe and measure entangled particles that are separated by large distances due to the redshifting of light. Additionally, the expansion of the universe can affect the probability density of entangled particles, as their distances from each other change over time.

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