# Expectation of Position of a Harmonic Oscillator

• Sekonda
You are correct in thinking that only the a-dagger operator increases the E-zero ket while the a operator acts as the annihilation operator. Therefore, the expectation value of position for the Harmonic Oscillator can be determined using the a-dagger operator in the position operator equation. In summary, the expectation value of position for the Harmonic Oscillator can be determined using the a-dagger operator in the position operator equation.
Sekonda
Hey,

My question is on determing the expectation value of position of the Harmonic Oscillator using raising and lowering operators, the question is part d) below:

I have determined the position operator to be:

$$\hat{x}=\sqrt{\frac{\hbar}{2m\omega}}(a+a^{\dagger})$$

and so the expectation value becomes:

$$\sqrt{\frac{\hbar}{2m\omega}}<E_{1}|(a+a^{\dagger})|E_{0}>$$

Though this is the bit I'm not sure about, I think only the a-dagger operator acts to increase the E-zero ket and the 'a' operator just causes the state to go to zero thus:

$$\sqrt{\frac{\hbar}{2m\omega}}=<E_{1}|\hat{x}|E_{0}>$$

Can someone confirm this?

Cheers,
SK

Looks good.

## 1. What is the expectation of position of a harmonic oscillator?

The expectation of position of a harmonic oscillator is the average or mean position that a particle would be found in if the experiment is repeated multiple times. It is a measure of the most probable location of the particle.

## 2. How is the expectation of position of a harmonic oscillator calculated?

The expectation of position of a harmonic oscillator can be calculated using the formula x = A sin(ωt + φ), where A is the amplitude, ω is the angular frequency, t is the time, and φ is the phase constant.

## 3. What factors affect the expectation of position of a harmonic oscillator?

The expectation of position of a harmonic oscillator is affected by the amplitude, frequency, and phase of the oscillation. It is also influenced by external factors such as temperature and potential energy.

## 4. Can the expectation of position of a harmonic oscillator be negative?

No, the expectation of position of a harmonic oscillator cannot be negative. This is because the position of a particle in a harmonic oscillator is always positive and the expectation is a measure of the most probable location.

## 5. How does the expectation of position of a harmonic oscillator relate to the uncertainty principle?

The expectation of position of a harmonic oscillator is related to the uncertainty principle in that the more accurately we know the position of a particle, the less certain we are about its momentum and vice versa. This is due to the wave-like nature of particles in a harmonic oscillator, where the more certain we are about the position, the less certain we are about the momentum and vice versa.

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