Question

3. (Total marks: 35) A spin 1/2 particle has the Hamiltonian $\hat{H}_0 = \hat{\sigma}_x B_x + \hat{\sigma}_y B_y$, where $(B_x, B_y) = (B_0 \cos \theta, B_0 \sin \theta)$, $B_0$ is real and positive, and $\hat{\sigma}_x = \begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix}$, $\hat{\sigma}_y = \begin{pmatrix} 0 & -i\\ i & 0 \end{pmatrix}$, $\hat{\sigma}_z = \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix}$ are Pauli matrices. (a) Find the energy eigenvalues and the corresponding normalized eigenfunctions of $\hat{H}_0$. 2 of 3 Then calculate the expectation values of $\hat{\sigma}_x$, $\hat{\sigma}_y$, $\hat{\sigma}_z$ for each eigenfunction. (8 marks) (b) If the spin system is prepared in the state $\chi = \begin{pmatrix} 1\\ 0 \end{pmatrix}$ at $t = 0$, please calculate the expectation values and uncertainty of $\hat{\sigma}_x$, $\hat{\sigma}_y$ for $t > 0$, and confirm that they satisfy the uncertainty principles. (15 marks) (c) If another term $\hat{H}_1 = \lambda \hat{\sigma}_x$ is applied to the spin particle, where $\lambda \ll B_0$, please use perturbation theory to find out the lowest non-vanishing order corrections to energy eigenvalues and eigenfunctions, and make a comparison with the exact solutions. (12 marks)

          3.
(Total marks: 35)
A spin 1/2 particle has the Hamiltonian $\hat{H}_0 = \hat{\sigma}_x B_x + \hat{\sigma}_y B_y$, where
$(B_x, B_y) = (B_0 \cos \theta, B_0 \sin \theta)$, $B_0$ is real and positive, and
$\hat{\sigma}_x = \begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix}$, $\hat{\sigma}_y = \begin{pmatrix} 0 & -i\\ i & 0 \end{pmatrix}$, $\hat{\sigma}_z = \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix}$ are Pauli matrices.
(a) Find the energy eigenvalues and the corresponding normalized eigenfunctions of $\hat{H}_0$.
2 of 3
Then calculate the expectation values of $\hat{\sigma}_x$, $\hat{\sigma}_y$, $\hat{\sigma}_z$ for each eigenfunction. (8 marks)
(b) If the spin system is prepared in the state $\chi = \begin{pmatrix} 1\\ 0 \end{pmatrix}$ at $t = 0$, please calculate the
expectation values and uncertainty of $\hat{\sigma}_x$, $\hat{\sigma}_y$ for $t > 0$, and confirm that they satisfy the
uncertainty principles. (15 marks)
(c) If another term $\hat{H}_1 = \lambda \hat{\sigma}_x$ is applied to the spin particle, where $\lambda \ll B_0$, please use
perturbation theory to find out the lowest non-vanishing order corrections to energy
eigenvalues and eigenfunctions, and make a comparison with the exact solutions. (12
marks)

3.
(Total marks: 35)
A spin 1/2 particle has the Hamiltonian Ĥ0 = σ̂x Bx + σ̂y By, where
(Bx, By) = (B0 cosθ, B0 sinθ), B0 is real and positive, and
σ̂x =
< p m a t r i x >, σ̂y =
< p m a t r i x >, σ̂z =
< p m a t r i x > are Pauli matrices.
(a) Find the energy eigenvalues and the corresponding normalized eigenfunctions of Ĥ0.
2 of 3
Then calculate the expectation values of σ̂x, σ̂y, σ̂z for each eigenfunction. (8 marks)
(b) If the spin system is prepared in the state χ =
< p m a t r i x > at t = 0, please calculate the
expectation values and uncertainty of σ̂x, σ̂y for t > 0, and confirm that they satisfy the
uncertainty principles. (15 marks)
(c) If another term Ĥ1 = λσ̂x is applied to the spin particle, where λ≪ B0, please use
perturbation theory to find out the lowest non-vanishing order corrections to energy
eigenvalues and eigenfunctions, and make a comparison with the exact solutions. (12
marks)

Added by James S.

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