# Quantum Potential Energy of SHM

• Mattofix
In summary, the conversation discusses a problem involving showing the potential energy of a particle in simple harmonic motion with frequency f. The equation for potential energy is given, but the speaker is having difficulty finding how to derive it. They mention that this topic is also covered in classical mechanics and ask about the relationship between spring force and potential energy.
Mattofix

## Homework Statement

I have been asked to show that a particle of mass m executing simple harmonic motion with frequency f, has a potential energy

V(x) = 2$$\pi^{2}$$f$$^{2}$$mx$$^{2}$$

## The Attempt at a Solution

I know that V(x) = 1/2m$$\omega^{2}$$x$$^{2}$$ and that $$\omega$$=2$$\pi$$f

but i can't find how to show this anywhere - not in the lecture notes, my books or the internet - I am finding this quantum mechanics course a massive step up from all the other areas of physics!

This is a problem that could be given to you in a classical mechanics course aswell.

What is the Spring-force (Hooks law), and how do you go from a (conservative)-Force to the potential?

## 1. What is quantum potential energy in SHM?

Quantum potential energy in SHM (simple harmonic motion) is the energy associated with the position of a particle in a quantum mechanical system. It is based on the principle of superposition, where the particle's position is described by a wave function that has both a potential energy and a kinetic energy component.

## 2. How does quantum potential energy affect SHM?

Quantum potential energy affects SHM by influencing the behavior of the particle in the system. As the particle moves back and forth, its position and energy are constantly changing due to the potential energy component of the wave function. This results in a quantized energy level for the particle, meaning it can only have certain discrete values of energy.

## 3. What is the relationship between quantum potential energy and classical potential energy in SHM?

In classical mechanics, potential energy is a continuous function of position, while in quantum mechanics, potential energy is a discrete function of position. This means that quantum potential energy differs from classical potential energy in that it has specific, quantized values rather than a continuous range of values.

## 4. How can quantum potential energy be calculated in SHM?

Quantum potential energy in SHM can be calculated using the Schrodinger equation, which describes the evolution of the wave function over time. By solving this equation, one can determine the potential energy component of the wave function and thus calculate the quantum potential energy at any given position.

## 5. Can quantum potential energy affect the amplitude of SHM?

Yes, quantum potential energy can affect the amplitude of SHM. As the particle's energy is quantized, it can only exist at certain energy levels. This means that the amplitude of the motion is limited to certain discrete values, and cannot have any arbitrary value like in classical mechanics.

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