# Time in QM vs SR: Energy, Frequency & Motion

• TEFLing
In summary: This means that the equation is not a simple Schrödinger equation, and thus cannot be interpreted in the same way as one. Additionally, the equation does not have a conserved probability density (like the Schrödinger equation does), which means that it cannot be interpreted as a probability amplitude in the same way.
TEFLing
In QM, the energy operator is proportional to the time derivative

E ~ d/dt

So higher energy particles have higher frequencies, i.e. their wave functions change more often per time than when at rest

But in SR, higher energy particles seem to exist in slow motion, appearing to age little

How are these theories compatible and reconcilable?

Do fast moving high energy particles change a lot per unit time as per QM

or freeze into slow motion as per SR ?

First of all, QM is not a relatiistic theory so you should not expect it to reproduce SR results. The extension of QM is relativistic QM and quantum field theory.

Second, your inference about SR is wrong. If you look at a highly relativistic particle in relativistic QM, it has a very high frequency compared to a particle at rest. If you want to make a classical analogy, you cannot forget about the particle momentum, which also will contribute to the phase, i.e., take into account that the particle is moving in space as well as in time.

So you're saying that we look at the plane wave solutions

Psi ~ exp( i( px - hwt )/h )
~ exp( i( hkx - hwt )/h )

w/k --> c

So if you were stationary, and the wave function flew by, you'd measure fast changing phase

But comoving with the wave, you would see no change in phase
The planes of constant phase would be speeding through space
At the velocity of the particle

So surfing those waves
As it were
One would see a frozen. Wave function
As it were

?

No. The classical QM (just as classical mechanics) ignores the main contribution to particle energy, its mass. Now this is fine in a classical theory since it is only an overall phase factor (or a constant energy shift if you will), but it is essential in relativistic QM in order to ensure Lorentz invariance.

Yes, this does not contradict what I just said.

TEFLing said:
comoving with the wave, you would see no change in phase

Would you? Try writing down a plane wave solution to the Klein-Gordon equation with the mass ##m## positive in a frame comoving with the wave. What do you get?

It cannot be straightforwardly interpreted as a Schrödinger equation for a quantum state, because it is second order in time and because it does not admit a positive definite conserved probability density.

What does the underlined mean exactly?

jerromyjon said:
What does the underlined mean exactly?

It means that the second time derivative appears instead of the first; ##\partial^2 \psi / \partial t^2## instead of ##\partial \psi / \partial t##.

## 1. How does quantum mechanics explain the concept of time?

Quantum mechanics explains time as a fundamental property of the universe, closely related to energy and motion. It is believed that time is continuous and can be described by the wave function, which is a mathematical representation of the state of a system. Time in quantum mechanics is also considered to be relative, meaning it is affected by the observer and the frame of reference.

## 2. How is time treated differently in quantum mechanics compared to special relativity?

In quantum mechanics, time is treated as a continuous variable that can be measured and described by the wave function. Special relativity, on the other hand, considers time to be a fourth dimension that is closely linked with space. It is treated as a relative concept, meaning it can be different for different observers depending on their relative speed and position.

## 3. How does energy relate to time in quantum mechanics and special relativity?

In quantum mechanics, the energy of a system is directly related to the frequency of its wave function. This means that as the energy of a system changes, so does the frequency of its wave function, which in turn affects the concept of time. In special relativity, energy is a component of the famous equation E=mc^2, which shows the relationship between energy and mass. This equation also has implications for the concept of time, as it shows that time can be affected by changes in energy and mass.

## 4. Can motion affect the passage of time in both quantum mechanics and special relativity?

Yes, both quantum mechanics and special relativity show that the passage of time can be affected by motion. In quantum mechanics, the speed and motion of particles can affect their wave function and therefore their concept of time. In special relativity, the relative motion of objects can cause time dilation, meaning that time can pass at different rates for different observers depending on their relative speed.

## 5. Is there a unified theory that can reconcile the concept of time in quantum mechanics and special relativity?

There is currently no unified theory that fully reconciles the concept of time in quantum mechanics and special relativity. However, some theories, such as quantum field theory, attempt to combine elements of both theories to describe the behavior of particles and the nature of time. Further research and experimentation are needed to fully understand the relationship between time, energy, and motion in these two theories.

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