Question about expectation values.

[cragar]

Is it possible to define operators to find the expectation value of position for a Gaussian wave packet. Similar to finding raising and lowering operators for the harmonic oscillator in terms of position and momentum and then using those to find <x> and <p>. But I was just wondering if this could be done for a Gaussian wave packet.

 

[mfb]

Here is one: X

If your wave packet is expressed as function of position, $<X> = \int \psi \psi^* x dx$.

 

[cragar]

I don't completely understand what you are doing? Is X my new operator.
are you starting with the definition of expectation value and then going from there.

 

[jtbell]

The position operator (in the position representation) is simply $x$. So the general definition of expectation value:

$$\langle A \rangle = \int {\Psi^* A_{op} \Psi dx}$$

becomes

$$\langle x \rangle = \int {\Psi^* x \Psi dx}$$

Plug in your wave function and grind out the integral.

 

[cragar]

ok i understand that. I was trying to think of a way to compute that without doing an integral. Like how you can do that for the harmonic oscillator with a+ and a -
like $<x>=<U|a^+ + a^-|U>$
u is the wave function and a+ is the raising operator.
can I do this for a Gaussian wave packet.

 

[mfb]

If you define appropriate operators.
For a gaussian wave packet, the expectation value of the position is the central value of the distribution. If you know that, you are done.