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Consider the Hamiltonian for a particle in a double well potential with both energies left and right $\bgroup\color{col1}$ \varepsilon_R=\varepsilon_L=0$\egroup$ and tunnel coupling $\bgroup\color{col1}$ T_c$\egroup$ ,

$\displaystyle H = \left(\begin{array}{cc} 0 & T_c \\ T_c & 0 \\ \end{array}\right).$

(6.1)

Consider an initial state at time $\bgroup\color{col1}$ t=0$\egroup$ ,

$\displaystyle \vert\Psi(t=0)\rangle = \vert L\rangle = \left(\begin{matrix}1 0 \end{matrix}\right).$

(6.2)

a) Calculate the state vector $\bgroup\color{col1}$ \vert\Psi(t)\rangle= \left(\begin{matrix}\alpha_L(t) \alpha_R(t) \end{matrix}\right)$\egroup$ for times $\bgroup\color{col1}$ t>0$\egroup$ .

b) Use the result from a) to calculate the probability to find the particle in the left well after time $\bgroup\color{col1}$ t$\egroup$ .

Tobias Brandes 2005-04-26