Expectation value of the time evolution operator

In summary, the problem pertains to the perturbative expansion of correlation functions in QFT. By using the fact that the interaction Hamiltonian satisfies i∂U/∂t = H_IU(t) and the properties of the vacuum state, it is shown that the expectation value of the time-ordered exponential of the interaction Hamiltonian is equal to the inverse of the expectation value of the complex conjugate of the same expression.
  • #1
Catria
152
4
This problem pertains to the perturbative expansion of correlation functions in QFT.

Homework Statement



Show that [itex]\langle0|T\left[exp\left(i\int_{-t}^{t}dt' H_{I}^{'}(t')\right)\right]|0\rangle = \left(\langle0|T\left[exp\left(-i\int_{-t}^{t}dt' H_{I}^{'}(t')\right)\right]|0\rangle\right)^{-1}[/itex]

Homework Equations



[itex]H_{I}[/itex] satisfies [itex]i \frac{\partial U}{\partial t} = H_{I}U(t)[/itex] and is the Hamiltonian in the interaction picture for the [itex]\varphi_{in}[/itex] fields, which was used in the perturbative expansion. Also, [itex]|0\rangle\langle0| = 1[/itex]

The Attempt at a Solution



[itex]1=\langle0|1|0\rangle = \langle0|T\left[exp\left((i-i)\int_{-t}^{t}dt' H_{I}^{'}(t')\right)\right]|0\rangle[/itex]

[itex]1=\langle0|T\left[exp\left(i\int_{-t}^{t}dt' H_{I}^{'}(t')\right)\right]|0\rangle\langle0|T\left[exp\left(-i\int_{-t}^{t}dt' H_{I}^{'}(t')\right)\right]|0\rangle[/itex]

From there it follows that [itex]\langle0|T\left[exp\left(i\int_{-t}^{t}dt' H_{I}^{'}(t')\right)\right]|0\rangle = \left(\langle0|T\left[exp\left(-i\int_{-t}^{t}dt' H_{I}^{'}(t')\right)\right]|0\rangle\right)^{-1}[/itex] simply by dividing...

I can't see what's wrong with my solution... but I have the feeling that there is something wrong.
 
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  • #2


Your approach seems correct. The only thing I would suggest is to be careful with the notation. The T in the first term is not necessary since you are taking the expectation value between vacuum states, so there is no need for time ordering. Also, it would be clearer to write the second term as the complex conjugate of the first term instead of (i-i).

So the final solution would be:

1 = <0|exp(i∫-t^t dt'H_I'(t'))|0> <0|exp(-i∫-t^t dt'H_I'(t'))|0>

1 = <0|exp(i∫-t^t dt'H_I'(t'))|0> * <0|exp(i∫-t^t dt'H_I'(t'))|0>*

<0|exp(i∫-t^t dt'H_I'(t'))|0> = (<0|exp(-i∫-t^t dt'H_I'(t'))|0>)^-1
 

1. What is the expectation value of the time evolution operator?

The expectation value of the time evolution operator is a mathematical quantity that represents the average value of an observable quantity over time. It is calculated by taking the inner product of the time evolution operator and the initial state of the system.

2. Why is the expectation value of the time evolution operator important?

The expectation value of the time evolution operator is important because it allows scientists to predict the behavior of a physical system over time. It is used in quantum mechanics to calculate the probabilities of different outcomes of measurements.

3. How is the expectation value of the time evolution operator calculated?

The expectation value of the time evolution operator is calculated by taking the inner product of the time evolution operator and the initial state of the system. This is then multiplied by the observable quantity being measured and integrated over time.

4. What factors can affect the expectation value of the time evolution operator?

The expectation value of the time evolution operator can be affected by factors such as the initial state of the system, the time period being considered, and the observable quantity being measured. It can also be influenced by external forces acting on the system.

5. How does the expectation value of the time evolution operator relate to uncertainty?

The expectation value of the time evolution operator is related to uncertainty through the Heisenberg uncertainty principle in quantum mechanics. This principle states that the more accurately we know the expectation value of one observable quantity, the less accurately we can know the expectation value of another observable quantity.

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