Question

Suppose a particle starts out in a linear combination ofjust two stationary states at t = 0: Ψ(x, 0) = c (x) + 2cψ2(x), where the eigen-energies for ψ1 and ψ2 are E1 and E2, respectively. a) Determine c b) What is the expectation value of the energy for the particle? c) What is the wave function Ψ(x, t) at subsequent times?

Answer 1

a)

c can be found out by normalization condition:

$\left \langle \psi \mid \psi \right \rangle = 1 \Rightarrow c^2[\psi_1^2(x) + 4\psi_2^2(x)] = c^2[1 + 4] = 5c^2 = 1$

$c = 1/\sqrt5$

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b)

Expectation value of the energy of the particle is:

$\left \langle \psi \mid \hat{H}\mid \psi \right \rangle = c^2[E_1\psi_1^2(x) + 4E_2\psi_2^2(x)] = \frac{1}{5}[E_1 + 4E_2]$

c)

Time evolution of the wavefunction is given as:

$\psi (x,t) = c[\psi_1(x) e^{-iE_1t/{\hbar}}+ 4\psi_2(x) e^{-iE_2t/{\hbar}}]$