Quantum expectation values

PhysicsForums.comIn summary, a user is having trouble with displaying their LaTeX code on PhysicsForums.com. They have attached a PDF of the question and have provided their attempt at a solution, but it seems to be incorrect. They are unsure if their solution is incorrect or if they are missing something in their problem. The forum user suggests double-checking the code for syntax errors and extending the calculation to include all three dimensions. They also recommend seeking assistance if needed.
  • #1
misterpickle
12
0
I'm not sure why PhysicsForums.com isn't displaying my latex properly so I have attached a PDF of the question.

Homework Statement


Show that, for a 3D wavepacket,

[itex] \frac{d\langle x^2 \rangle}{dt} $=$ \frac{1}{m}(\langle xp_{x} \rangle+\langle p_{x}x \rangle) [/itex]

The Attempt at a Solution


Through several pages of algebra and taking divergences I have come to the following.

[itex] \frac{d\langle x^2\rangle}{dt} $=$ \frac{ih}{2m}\int\int\int{x^{2}(\psi\frac{\partial^{2}{\psi^{*}}}{\partial{x^{2}}}-\psi^{*}\frac{\partial^{2}{\psi}}{\partial{x^{2}}})+4x(\psi\frac{\partial{\psi^{*}}}{\partial{x}}}-\psi^{*}\frac{\partial{\psi}}{\partial{x}})\,dx\,dy\,dz [/itex]

I know the values of <x> and <p> and that my equation must, somehow (if correct), equate to the problem statement. I'm not quite sure if I just don't know how to finish this problem or if my solution thus far is incorrect.
 

Attachments

  • quantum5.pdf
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  • #2




Thank you for sharing your question with us. It seems like your LaTeX code may not be displaying properly due to some syntax errors. I would suggest double-checking your code and making sure all commands are properly closed and that you are using the correct symbols for operators and variables.

As for your solution, it is important to note that the equation you have derived is for a 1D wavepacket, not a 3D one. In order to obtain the correct equation for a 3D wavepacket, you will need to extend your calculation to include the y and z directions as well. This will involve taking partial derivatives with respect to y and z and integrating over all three dimensions.

Once you have obtained the correct 3D equation, you can substitute in the values for <x> and <p> to see if it matches the problem statement. If you are still having trouble, I would recommend seeking assistance from a colleague or your professor to ensure your solution is correct.

I hope this helps and good luck with your studies.



 

1. What are quantum expectation values?

Quantum expectation values are a mathematical concept used to describe the average value of a physical quantity in quantum mechanics. They are obtained by taking the inner product of a quantum state with the operator representing the physical quantity.

2. How are quantum expectation values calculated?

To calculate a quantum expectation value, we first need to determine the quantum state of the system and the operator representing the physical quantity of interest. We then take the inner product of these two quantities and perform any necessary mathematical operations to obtain the final value.

3. What is the significance of quantum expectation values?

Quantum expectation values are significant because they allow us to make predictions about the behavior of quantum systems. They provide us with information about the average values of physical quantities, which can help us understand the behavior of particles at the quantum level.

4. How do quantum expectation values differ from classical expectation values?

Quantum expectation values differ from classical expectation values in that they take into account the probabilistic nature of quantum mechanics. In classical mechanics, the value of a physical quantity is known with certainty, whereas in quantum mechanics, it is described by a probability distribution.

5. Can quantum expectation values be measured?

Yes, quantum expectation values can be measured through repeated experiments on identical quantum systems. By performing measurements on a large number of identical systems, we can obtain an average value that corresponds to the quantum expectation value.

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