Quantum Mechanics: Expectation values

In summary, the conversation is about finding the expectation value for E, which is represented by the equation b = \int dt \phi^{*}(t) \hat{{\cal E}}_{in}(t). The problem is not fully understood and clarification is needed on the system, operators, and variables involved. The goal is to compute the mean value and standard deviation of the single photon amplitude, using the operator \hat{{\cal E}} = e^{-\kappa \tau}+ e^{-\kappa t}\int^{t}_{0}e^{\kappa \tau} \sqrt{2\kappa}\, \hat{{\cal E}}_{in}(\tau)dt in Heisen
  • #1
Nusc
760
2

Homework Statement



I need to find the expectation value for E but I don't know how b acts on the vacuum state.

Homework Equations


[tex]
b = \int dt \phi^{*}(t) \hat{{\cal E}}_{in}(t)
[/tex]
[tex]
| \psi(t)\rangle = b^\dagger| 0\rangle
[/tex]



The Attempt at a Solution


[tex]
\langle \psi(t) | \hat{{\cal E}}^\dagger\hat{{\cal E}}| \psi(t)\rangle =

[/tex]
 
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  • #2
What did I not make clear?
 
  • #3
Nusc said:
What did I not make clear?

Everything! What is the system supposed to be? How are your operators and other variables defined? Are you integrating over time? Then how is it that the state is time dependent? What exactly are you supposed to compute the expectation value of?
 
  • #4
This represents the single photon output level and I'm supposed to determine the
mean value and standard deviation of the single photon amplitude.
[tex]
\hat{{\cal E}} = e^{-\kappa \tau}+ e^{-\kappa t}\int^{t}_{0}e^{\kappa \tau} \sqrt{2\kappa}\, \hat{{\cal E}}_{in}(\tau)dt
[/tex]

I'm integrating with respect to time.

[tex]
{\cal E}
[/tex]is an operator in Heisenberg picture.

b^+ creates a photon in the temporal mode [tex]\phi(t)[/tex]

Does that make sense?
 
  • #5
Sorry, it does not make sense to me. Perhaps someone else will be able to help.
 

1. What is the concept of expectation values in quantum mechanics?

The concept of expectation values in quantum mechanics is a mathematical tool used to calculate the average value of a physical quantity in a quantum system. It is calculated by taking the sum of all possible outcomes of the quantity multiplied by their respective probabilities.

2. How is expectation value related to uncertainty in quantum mechanics?

In quantum mechanics, the uncertainty principle states that it is impossible to know the exact position and momentum of a particle simultaneously. The expectation value of these quantities can give an estimate of their average values, but there will always be some degree of uncertainty involved.

3. What is the difference between expectation value and measurement in quantum mechanics?

Expectation value is a theoretical calculation that gives the average value of a physical quantity in a quantum system, while measurement is the actual act of observing and obtaining a specific value for that quantity. The measurement of a quantity in quantum mechanics can also affect the state of the system, while the expectation value does not.

4. How can expectation values be used to describe the behavior of quantum particles?

Expectation values can be used to describe the behavior of quantum particles by providing information about the average values of their physical quantities. This can help predict how a particle will behave and interact with other particles in a quantum system.

5. Can expectation values be negative in quantum mechanics?

Yes, expectation values can be negative in quantum mechanics. This can occur when a physical quantity has a greater probability of having a negative value in a given quantum system. However, expectation values are not always physically meaningful and should be interpreted with caution.

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