Why do we consider the evolution (usually in time) of a wave function ?

In summary, the evolution of a wave function is considered in quantum mechanics because it allows us to understand what will happen next in a system. The evolution parameter is typically taken as time because it is the most convenient and relevant parameter for experiments in a laboratory. This is also supported by Pauli's argument that taking time as an observable would lead to contradictions in the stability of matter.
  • #1
rajesh_d
6
1
Why do we consider evolution of a wave function and why is the evolution parameter taken as time, in QM.

Look at a simple wave function $\psi(x,t) = e^{kx - \omega t}$. $x$ is a point in configuration space and $t$ is the evolution parameter. They both look the same in the equation, then why consider one as an evolution parameter and other as configuration of the system.

My question is why should we even consider the evolution of the wave function in some parameter (it is usually time)?. Why can't we just deal with $\psi(\boldsymbol{x})$, where $\boldsymbol{x}$ is the configuration of the system and that $|\psi(\boldsymbol{x})|^2$ gives the probability of finding the system in the configuration $\boldsymbol{x}$?

One may say, "How to deal with systems that vary with time?", and the answer could be, "consider time also as a part of the configuration space". Why wonder why this could not be possible.
 
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  • #2
Why do we consider evolution of a wave function and why is the evolution parameter taken as time, in QM.
Because we want to know what happens next. Generally a system will change in time... so the square modulus of the wavefunction only give the correct probability-density at a particular point in time.

Yes we can work time into the configuration space - this is the point of relativistic quantum mechanics.

Even so, we have to do experiments in a laboratory which is mired in time - so the coordinate that is usually most convenient to evolve the wavefunction with respect to is still time.
 
  • #3
Time in quantum theory is necessarily a parameter labelling the "causal sequence of events" or something like that, no matter whether you consider non-relativistic or relativistic quantum theory.

The reason has been given by Pauli very early in the development of quantum theory: If time would be taken as an observable than it would have the commutation relation
[tex][\hat{t},\hat{H}]=\mathrm{i}\hbar[/tex].
This would mean, as for the case of the position and momentum operator in non-relativistic quantum theory, that the spectrum of both the time and the Hamilton operator would be whole [itex]\mathbb{R}[/itex], and this contradicts the stability of matter since for that the energy must have a lower boundary, so that a ground state of lowest energy exists.
 

1. Why is it important to consider the evolution of a wave function?

The evolution of a wave function is important because it allows us to understand how a system changes over time. By studying the changes in the wave function, we can gain insight into the behavior and properties of the system, and make predictions about its future states.

2. What factors influence the evolution of a wave function?

The evolution of a wave function is influenced by a variety of factors, including the initial conditions of the system, the interactions between particles, and external forces acting on the system. These factors can cause the wave function to change in shape, amplitude, and frequency over time.

3. How does the evolution of a wave function relate to the uncertainty principle?

The uncertainty principle states that it is impossible to simultaneously know the exact position and momentum of a particle. The evolution of a wave function is directly related to this principle, as it describes the probabilistic nature of particles and their behavior. The wave function allows us to calculate the probability of finding a particle in a certain state, rather than determining its exact position and momentum.

4. Can the evolution of a wave function be observed?

The evolution of a wave function cannot be directly observed, as it is a mathematical description of the behavior of particles. However, we can indirectly observe its effects through experiments and measurements. For example, by studying the interference patterns of particles, we can gain information about the changes in their wave functions.

5. How does the evolution of a wave function differ from classical mechanics?

In classical mechanics, the state of a system can be precisely determined at any given time. However, in quantum mechanics, the behavior of particles is described by the wave function, which is probabilistic in nature. This means that the evolution of a wave function differs from classical mechanics, as it describes the likelihood of a particle being in a certain state rather than its exact state.

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