Which of these interpretations of the modulus squared of wavefunction is right?

In summary, the probability of a particle to collapse at a specific point in space at a specific time is given by the integral of the squared wave function over that point in space, not over time. This is according to Born's interpretation in nonrelativistic quantum mechanics. However, this does not directly predict the frequency of counts in a Geiger counter, as it only gives the probability of the particle being at a certain point in space, not the probability of being detected at that point. The number of counts in a Geiger counter is proportional to the time interval of the measurement, but each individual particle can be described by a normalized wave function that gives the probability density in space.
  • #1
dEdt
288
2
Does [itex]|\psi(\mathbf{x},t)|^2d^3\mathbf{x}[/itex] or [itex]|\psi(\mathbf{x},t)|^2d^3\mathbf{x}dt[/itex] give the probability of a particle to collapse at the point [itex]\mathbf{x}[/itex] at time [itex]t[/itex]?

Griffiths sides with the former, but I'm having doubts.
 
Physics news on Phys.org
  • #2
dEdt said:
Does [itex]|\psi(\mathbf{x},t)|^2d^3\mathbf{x}[/itex] or [itex]|\psi(\mathbf{x},t)|^2d^3\mathbf{x}dt[/itex] give the probability of a particle to collapse at the point [itex]\mathbf{x}[/itex] at time [itex]t[/itex]?

Griffiths sides with the former, but I'm having doubts.

It's the former, it doesn't make sense to integrate over time, at any instant t, the integration over space gives you the overall probability at that time, which is 1.
 
  • #3
cattlecattle said:
It's the former, it doesn't make sense to integrate over time, at any instant t, the integration over space gives you the overall probability at that time, which is 1.

Here are my issues: 1) space and time seem to be treated on different footings, and 2) the probability that say a Geiger counter goes off at a given time t is zero, but the probability that it goes off over some time interval is non-zero.
 
  • #4
dEdt said:
Here are my issues: 1) space and time seem to be treated on different footings,

Yes. This is nonrelativistic quantum mechanics, which treats space and time differently. To fix this we have relativistic quantum field theory.

dEdt said:
2) the probability that say a Geiger counter goes off at a given time t is zero, but the probability that it goes off over some time interval is non-zero.

If you do an ideal position measurement at time t, the probability of finding the particle *somwhere* is 1. Geiger counters don't do ideal position measurements; the quantum mechanical analysis of radioactive decay is somewhat more complicated.
 
  • #5
2) the probability that say a Geiger counter goes off at a given time t is zero, but the probability that it goes off over some time interval is non-zero.

This is true, but I think the probability given by [itex]|\psi^2|dV[/itex] according to Born's interpretation should not be used directly to predict the frequency of counts of the Geiger detector (unless one smuggles the source intensity to [itex]\psi[/itex], which can allow us to do just that; but then the above form of Born's rule is not applicable.)

Instead, if the wave function for a particle is normalized (the most clear approach), it gives us the probability that this particle is at some point of space (without the necessity to detect it there).

You are right that the number of counts(clicks) of detector set in some definite distance from the piece of matter scattering charged particles will be proportional to time interval of the measurement, but this is because larger interval allows more [itex]\textit{distinct particles}[/itex] to come at the detector; however, each one can be ascribed by normalized wave function that gives density of probability in space by the rule [itex]|\psi|^2dV[/itex].
 

1. What is the modulus squared of a wavefunction?

The modulus squared of a wavefunction is a mathematical expression that describes the probability of finding a particle in a particular state. It is calculated by squaring the absolute value of the wavefunction.

2. Why is the modulus squared of a wavefunction used in quantum mechanics?

In quantum mechanics, the wavefunction represents the state of a particle. However, the wavefunction itself is a complex-valued function, making it difficult to interpret physically. Therefore, the modulus squared is used to find the probability of finding the particle in a particular state, which can be directly measured.

3. What are the different interpretations of the modulus squared of a wavefunction?

There are two main interpretations of the modulus squared of a wavefunction: the Copenhagen interpretation and the Many-Worlds interpretation. The Copenhagen interpretation states that the wavefunction collapses into a single state upon measurement, while the Many-Worlds interpretation suggests that all possible states of the particle exist simultaneously in different universes.

4. Which interpretation of the modulus squared of a wavefunction is considered "right"?

There is currently no consensus on which interpretation is considered "right" in the scientific community. The debate between the Copenhagen and Many-Worlds interpretations is ongoing and has not been definitively resolved.

5. Can the modulus squared of a wavefunction be directly observed or measured?

No, the modulus squared of a wavefunction cannot be directly observed or measured. It is a mathematical concept that represents the probability of finding a particle in a particular state. However, experiments can be designed to indirectly measure the modulus squared through measurements of the particle's properties.

Similar threads

  • Quantum Interpretations and Foundations
Replies
5
Views
2K
  • High Energy, Nuclear, Particle Physics
Replies
3
Views
845
  • Quantum Interpretations and Foundations
Replies
14
Views
2K
  • Quantum Interpretations and Foundations
9
Replies
309
Views
8K
  • Differential Equations
Replies
2
Views
1K
  • Quantum Interpretations and Foundations
3
Replies
84
Views
1K
Replies
2
Views
667
Replies
2
Views
1K
  • Quantum Interpretations and Foundations
Replies
17
Views
1K
  • Differential Equations
Replies
1
Views
624
Back
Top