- #1
bgrape
- 2
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Homework Statement
A two state system has the following hamiltonian
[tex]
H=E \left( \begin{array}{cc} 0 & 1 \\
1 & 0 \end{array} \right)
[/tex]
The state at t = 0 is given to be
[tex]
\psi(0)=\left( \begin{array}{cc} 0 \\ 1 \end{array} \right)
[/tex]
• Find Ψ(t).
• What is the probability to observe the ground state energy at t = T ?
• What is the probability that the state is
[tex]
\left( \begin{array}{cc} 1 \\ 0
\end{array} \right)
[/tex]
at t = T ?
Homework Equations
[tex]
i\hbar\frac{\partial\psi}{\partial t} = \frac{\hbar^2}{2m}\nabla^2\psi + V(\mathbf{r})\psi
[/tex]
The Attempt at a Solution
My main problem is with the notation. I understand that this system has two states with the same energies. I think this system can be an electron in an energy level, it has up or down spin possibilities. I think since both the states have the same energy both are considered ground state therefore we have 100% possibility to observe the ground state energy at t=T and the probability that the state is [tex]
\left( \begin{array}{cc} 1 \\ 0
\end{array} \right)
[/tex] at t=T seems to be 1/2 intuitively because there is no energy difference between the states. However I don't know the correct notation for Ψ(t). I am thinking of
[tex]
\psi(t)=\left( \begin{array}{cc} e^{(-iEt/\hbar)} \\
e^{(-iEt/\hbar)} \end{array} \right)
[/tex]
but this does not define that the system is in the particular state at t=0. Any help is greatly appriciated.