# Expectation value of momentum in symmetric 2D H.O

• ma18
In summary, the problem involves calculating the expectation value of the momentum operator for two initial states of the symmetric 2D harmonic oscillator, given by ket (phi 1) and ket (phi 2). The expectation value is represented by <phi_1 | p_x (t) | phi_2>, where p_x (t) is the momentum operator. To solve this, one must first find the general solution for the 2D harmonic oscillator and then use it to calculate the expectation value.
ma18

## Homework Statement

Consider the following inital states of the symmetric 2D harmonic oscillator

ket (phi 1) = 1/sqrt(2) (ket(0)_x ket(1)_y + ket (1)_x ket (0)_y)

ket (phi 2) = 1/sqrt(2) (ket(0)_x ket(0)_y + ket (1)_x ket (0)_y)

Calculate the <p_x (t)> for each state

## The Attempt at a Solution

I am not sure how to work with these kets, I know that for the expectation value using you would do
<phi_1 | p_x (t) | phi_2>

but I don't know how to represent p_x (t) in the notation used in the ketsAny help would be much appreciated.

Last edited:
The expectation value for an operator on a state is <state 1| operator |state 1>. You wrote <state 1| operator |state 2>. Note the 2 in the last ket. That's a transition amplitude as induced by the operator.

Look in your text for a representation of the momentum operator. You should find it near he Heisenberg uncertainty formula, or the Schrodinger equation or some such. You probably want a derivative with some constants.

Notice that the problem says "initial states." So these are not the wave functions. These are the t=0 values. It does not give the x dependence, nor the t dependence. You will need to do some reading in your text to find the 2-D harmonic oscillator and the general solution for it.

## What is the expectation value of momentum in a symmetric 2D harmonic oscillator?

The expectation value of momentum in a symmetric 2D harmonic oscillator is a measure of the average momentum of a particle in the system. It is given by the integral of the momentum operator over the wavefunction, and can be calculated using the Schrödinger equation.

## How is the expectation value of momentum related to the uncertainty principle?

The expectation value of momentum is related to the uncertainty principle through the Heisenberg uncertainty principle, which states that the product of the uncertainties in position and momentum must be greater than or equal to a constant value. This means that a more precise knowledge of the momentum of a particle will result in a less precise knowledge of its position, and vice versa.

## What factors affect the expectation value of momentum in a symmetric 2D harmonic oscillator?

The expectation value of momentum in a symmetric 2D harmonic oscillator is affected by the mass of the particle, the frequency of oscillation, and the shape and size of the potential well. These factors can change the energy levels and wavefunction of the particle, which in turn affects the probability of measuring a certain momentum.

## How can the expectation value of momentum be experimentally determined in a symmetric 2D harmonic oscillator?

The expectation value of momentum can be experimentally determined by measuring the position of a particle at different points in time and calculating its average momentum using classical mechanics. It can also be directly measured using advanced techniques such as quantum tomography, which allows for the reconstruction of the wavefunction and its associated momentum distribution.

## What is the significance of the expectation value of momentum in studying the behavior of particles in a symmetric 2D harmonic oscillator?

The expectation value of momentum is a fundamental quantity in quantum mechanics and is used to study the behavior of particles in a symmetric 2D harmonic oscillator. It provides information about the average momentum of a particle and its spread in momentum space, which can help determine the probability of finding the particle in a certain state. It also plays a crucial role in understanding the dynamics and energy levels of the system.

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