Calculation of a Propagator for a Spin 1/2 system

[TheShadowDragon]

Homework Statement

Evaluate the expression below.
The $S_{x}(t_i)$ are the usual Spin 1/2 Matrices. T is the time ordered product. The Hamiltonian is $\omega_0 \sigma_z+\lambda \sigma_x$. The ground state when $\lambda=0$ is $\ket{0}$ and when it is non zero, the ground state is $\ket{\omega}$. $H_I=\lambda \sigma_x$ is the Interaction Hamiltonian in the Interaction picture.

Relevant Equations

$$\bra{\omega}T[S_x(t_1)S_x(t_2)]\ket{\omega}=\lim_{T\to\infty(1-i\epsilon)}\frac{\bra{0}T[S_x(t_1)S_x(t_2)exp(-i\int_{-T}^{T}H_{I}(t)dt\ket{0}}{\bra{0}T[exp(-i\int_{-T}^{T}H_{I}(t)dt\ket{0}}$$
Note that, on the right hand side of this equation, the spin matrices are in the Interaction picture and on the left hand side, they are in the Heisenberg picture.

Well, this calculation is straightforward in the Heisenberg picture. After finding the eigen values and eigen vectors of the total Hamiltonian, I found the explicit form for the exponential of the integral of the matrix and then did the matrix multiplication and calculated its expectation value in the new ground state.
Now, my problem is in the calculation using the Dyson Formula. If I explicitly calculate the exponential of the matrix and then apply the Wick contraction theorem, I end up with the free propagator since the terms coming out of the exponential in the numerator and denominator cancel out.
I would be grateful for any advice on how to proceed in this problem.

 

[mark726]

Thank you for sharing your approach to the problem in the Heisenberg picture. It seems like you have made good progress in finding the ground state expectation value using the eigenvalues and eigenvectors of the total Hamiltonian.

In terms of using the Dyson Formula, it is important to make sure that you are applying it correctly. The Wick contraction theorem is a useful tool for simplifying the calculation, but it is important to remember that it only applies to certain types of operators. It may be helpful to review the assumptions and limitations of the Wick contraction theorem and make sure that your calculation satisfies them.

Additionally, it may be helpful to consider alternative approaches to the problem. Have you explored other methods, such as the path integral formulation or the interaction picture? Sometimes, a different perspective can provide insight into a problem and lead to a solution.

I hope this advice is helpful and I wish you success in your calculations.