# QM: expectation value of a harmonic oscillator

• ktravelet
In summary, the conversation is about finding <x> and <p> for the nth stationary state of the harmonic potential V(x)=(1/2)mw^2x^2. The speaker solved for x and found that both <x> and <p> are equal to zero. They are asking for help finding <T>.
ktravelet
first post! but for bad reasons lol

Im trying to find <x> and <p> for the nth stationary state of the harmonic potential: V(x)=(1/2)mw^2x^2

i solved for x: x=sqrt(h/2mw)((a+)+(a-))
so <x> integral of si x ((a+)+(a-)) x si.
therefor the integral of si(n+1) x si + si(n-1) x si.
si(n+1) x si as far as I know is always 0 so this would mean <x>=0?
this same convention would be used for <p>.

sorry for the sloppy work but I'm pretty sure <x>, and <p> shouldn't = 0

ktravelet said:
first post! but for bad reasons lol

Im trying to find <x> and <p> for the nth stationary state of the harmonic potential: V(x)=(1/2)mw^2x^2

i solved for x: x=sqrt(h/2mw)((a+)+(a-))
so <x> integral of si x ((a+)+(a-)) x si.
therefor the integral of si(n+1) x si + si(n-1) x si.
si(n+1) x si as far as I know is always 0 so this would mean <x>=0?
this same convention would be used for <p>.

sorry for the sloppy work but I'm pretty sure <x>, and <p> shouldn't = 0

welcome!

Your result is correct! They are both zero. You can see it directly if you think in terms of raising and lowering operator since, for any eigenstate |n>, we have

$$\langle n| a | n\rangle = \langle n | a^\dagger | n \rangle = 0$$

Alright thanks a lot! I have absolutely no idea how to find <T>. If you can give me a pointer in the right direction that would be greatly appreciated. Thank you very much for all the help.

## 1. What is the expectation value of a harmonic oscillator in quantum mechanics?

The expectation value of a harmonic oscillator in quantum mechanics is the average value that would be obtained if the system were measured multiple times. It is calculated by taking the integral of the wave function squared multiplied by the operator for the observable quantity.

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

The expectation value of a harmonic oscillator is calculated by taking the integral of the wave function squared multiplied by the operator for the observable quantity. This integral is then divided by the normalization constant of the wave function.

## 3. What is the significance of the expectation value in quantum mechanics?

The expectation value in quantum mechanics is significant because it represents the most likely outcome of a measurement for a given observable quantity. It also provides information about the probability distribution of the system.

## 4. How does the expectation value change with different states of a harmonic oscillator?

The expectation value of a harmonic oscillator changes with different states because the wave function and the operator for the observable quantity will be different for each state. This leads to different integrals and ultimately different expectation values.

## 5. How is the expectation value of a harmonic oscillator related to the uncertainty principle?

The expectation value of a harmonic oscillator is related to the uncertainty principle in that it can be used to calculate the standard deviation of the system. The uncertainty principle states that the product of the standard deviations of two complementary observables must be greater than or equal to the absolute value of their commutator. This means that the more precise the expectation value is, the larger the uncertainty of the system will be.

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