# Quantum oscillator from position to momentum space

• Dazed&Confused
In summary, the conversation discusses the substitution of ##x\to p## and ##m\omega\to 1/m\omega## to obtain the corresponding wave function of a quantum harmonic oscillator in momentum space from position space. However, there seems to be a mistake when deriving the TISE for momentum space. The correct operators are ##p## and ##i\hbar d/dp##, and the equivalent equation in position space is given. The link provided may be helpful in understanding this concept further.
Dazed&Confused
So I've read you can get the corresponding wave function of a quantum harmonic oscillator in momentum space from position space by making the substitution ##x \to k## and ##m \omega \to 1/m \omega##.

However in deriving the TISE for momentum space, I seem to be making a mistake. In momentum space ##P## acts like ##\hbar k## and ##X## acts like ##i d/dk.##

The TISE is $$\left ( \frac{P^2}{2m} + \frac12 m \omega^2 X^2 \right) \psi = \left ( \frac{\hbar^2 k^2}{2m} - \frac12 m \omega^2 \frac{d^2}{dk^2} \right ) \psi = E\psi.$$

Therefore $$\psi'' + \frac{2}{m \omega^2} \left ( E - \frac{\hbar^2 k^2}{2m} \right) = 0.$$

The equivalent equation in position space is $$\psi'' + \frac{2m}{\hbar^2} \left ( E -\frac12 m\omega^2x^2 \right ) \psi = 0.$$

I'm not sure what I'm doing wrong. Edit: incorrect operators, can be deleted.

Last edited:
It's not ##x\to k##, it's ##x\to p##. So replace ##P## with ##p## and ##X## with ##i\hbar d/dp##, and see what you get.

## What is a quantum oscillator and how does it relate to position and momentum space?

A quantum oscillator is a mathematical representation of a system with a finite number of discrete energy levels. It can be used to describe the behavior of particles in a potential well, such as an atom in a molecule. The oscillator is related to position and momentum space through the Heisenberg uncertainty principle, which states that the more precisely we know the position of a particle, the less precisely we can know its momentum, and vice versa.

## What is the difference between position space and momentum space in quantum mechanics?

Position space in quantum mechanics refers to the representation of a system in terms of the position of its particles. Momentum space, on the other hand, refers to the representation of a system in terms of the momentum of its particles. These two spaces are related through a mathematical transformation, known as a Fourier transform, which allows us to switch between the two representations.

## How is the quantum oscillator represented in position and momentum space?

The quantum oscillator is represented in position space by a wave function, which describes the probability of finding the particle at a given position. In momentum space, the oscillator is represented by a Fourier transform of the wave function, which describes the probability of finding the particle with a given momentum.

## What is the significance of the uncertainty principle in the quantum oscillator?

The uncertainty principle plays a crucial role in the quantum oscillator as it dictates the relationship between the position and momentum representations. It tells us that the more precisely we know the position of the oscillator, the less precisely we can know its momentum, and vice versa. This has important implications for the behavior of particles in the oscillator, as it introduces inherent uncertainty and randomness into their properties.

## How is the quantum oscillator used in practical applications?

The quantum oscillator has a wide range of applications in various fields such as quantum mechanics, quantum computing, and quantum optics. It is used to model and understand the behavior of particles in different systems, and its mathematical representation is also used in developing algorithms for quantum computers. In quantum optics, it is used to describe the behavior of light in optical cavities and resonators. Additionally, the quantum oscillator has been used in the development of new technologies, such as lasers and atomic clocks.

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