Problem 2.1ΒΆ

Prove the following three theorems:

  1. For normalizable solutions, the separation constant E must be real. Hint: Write \(E\) (in Equation 2.7) as \(E_0+i\Gamma\) (with \(E_0\) and \(\Gamma\)), and show that if Equation 1.20 is to hold for all \(t\), \(\Gamma\) must be zero.

  2. The time-independent wave function \(\psi(x)\) can always be taken to be real (unlike \(\Psi(x,t)\), which is necessarily complex).

Solution:

  1. Substituting \(E+i\Gamma\) into Equation 2.7 gives

\[\psi(x,t) = \psi(x) e^{-i(E_0+i\Gamma)t/\hbar} = \psi(x) e^{-iE_0t/\hbar} e^{-\Gamma t/\hbar}\]

Taking the modulus squared and integrating over all space, we have

\[\int_{-\infty}^{\infty} |\psi(x,t)|^2 dx = e^{-2\Gamma t/\hbar} \int_{-\infty}^{\infty} |\psi(x)|^2 dx=1,\]

which implies that

\[e^{-2\Gamma t/\hbar} = 1 \rightarrow \text{apply logarithm}\rightarrow -2\Gamma t/\hbar = 0.\]

Since this must hold for all \(t\), \(\hbar\) and 2 are constants, we conclude that \(\Gamma = 0\), and hence \(E\) is real.