An equation about local conservation in quantum mechanics

However, when considering vibration, we must also take into account the complex amplitude and the rule of local conservation. In summary, a particle's wave function can be described as a superposition of sine waves or as a vibration, but these two explanations are not equal. To calculate momentum, we can use either the concept of wave superposition and Fourier transform, or the equation $$P = mv$$ while also considering the complex amplitude and the rule of local conservation.
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Homework Statement


Like ordinary wave, a particle’s wave function can be described as countless line of sine wave’s superposition. However it can also be clarified as vibration (the complex amplitude still follow the rule of local conservation) I think these two explanation are equal. Am I wrong? You can regard it as wave superposition and use Fourier transform to calculate the momentum P. However, if you regard it as vibration, how to do it?

Homework Equations


The Attempt at a Solution

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This's the first time I have used this forum.There are some complex equations in the description of the question.So I use an image.The main description about the question is in the image.
 

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No, the two explanations are not equal. Wave superposition is the combination of multiple waves that can be described using Fourier transform. Vibration, on the other hand, is a periodic oscillation or motion about a fixed point. To calculate momentum P, we use the equation $$P = mv$$ where m is the mass of the particle and v is the velocity.
 

Related to An equation about local conservation in quantum mechanics

1. What is local conservation in quantum mechanics?

Local conservation in quantum mechanics refers to the principle that within a small region of space, energy, momentum, and other physical quantities are conserved. This means that the total amount of these quantities in a given area will not change over time, even as particles and energy interact within that region.

2. How is local conservation expressed in equations?

The most commonly used equation to express local conservation in quantum mechanics is the continuity equation, which states that the change in a physical quantity within a given volume is equal to the flux of that quantity across the boundary of that volume. This equation is often written in terms of operators and wave functions to describe the behavior of quantum particles.

3. What is the significance of local conservation in quantum mechanics?

Local conservation is a fundamental principle in quantum mechanics that helps to explain the behavior of particles and energy at a small scale. It allows us to make predictions about how these quantities will behave within a given system, and is an essential concept in fields such as atomic and molecular physics, condensed matter physics, and quantum field theory.

4. Are there any exceptions to local conservation in quantum mechanics?

While local conservation is a general principle in quantum mechanics, there are certain situations where it may not hold true. For example, in systems with strong interactions or high energies, local conservation may break down. Additionally, in quantum systems that are not in equilibrium, local conservation may not be applicable.

5. How does local conservation relate to other principles in quantum mechanics?

Local conservation is closely related to other principles in quantum mechanics, such as the conservation of total energy and momentum. It also ties into the concept of gauge symmetry, which is a fundamental symmetry in quantum mechanics that allows us to make predictions about the behavior of particles and energy. Local conservation can also be seen as a consequence of the laws of quantum mechanics, which govern the behavior of particles at a small scale.

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