
#1
Dec1112, 09:12 AM

P: 2,453

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.




#2
Dec1112, 11:03 AM

Mentor
P: 10,766

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



#3
Dec1212, 08:35 AM

P: 2,453

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. 



#4
Dec1212, 08:46 AM

Mentor
P: 11,217

question about expectation values.
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. 



#5
Dec1312, 08:05 PM

P: 2,453

ok i understand that. I was trying to think of a way to compute that with out doing an integral. Like how you can do that for the harmonic oscillator with a+ and a 
like [itex] <x>=<Ua^+ + a^U> [/itex] u is the wave function and a+ is the raising operator. can I do this for a Gaussian wave packet. 



#6
Dec1412, 08:11 AM

Mentor
P: 10,766

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. 


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