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Time evolution of a superposition of two energy eigenstates

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The Schrödinger equation is:

i hbar d|psi>/dt = H|psi> (1)

The initial state is:

|psi(t=0)> = N(|1> + |2>)

where |1> and |2> are two eigenstates of the Hamiltonian:

H|1> = E1 |1>

H|2> = E2 |2>

And E1 is not equal to E2 (by definition as the two states are nondegenerate)

The normalization constant N = 1/sqrt(2) is |1> and |2> are normalized.

We can solve Eq. (1) as follows. Since the equation is a linear differential equation, a superposition of two solutions is another solution. Such a superposition will satisfy the superposition of the initial conditions of the individual solutions. Now, we can write any state as a superposition of eigenstates of the Hamiltonian. So, all we need to do is to solve Eq. (1) for the initial condition were ...

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A detailed solution is given regarding time evolution of a superposition of two energy eigenstates.

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