2.4.2 Statistical selection

1

Suppose the wave function is initially

$$\Psi = \frac{1}{\sqrt{2}} \psi_1 + \frac{1}{\sqrt{2}} \psi_2$$

where $\psi_1$ is an normalized eigenfunction of the Hamiltonian with eigenvalue $E_1$, and $\psi_2$ an eigenfunction with eigenvalue $E_2$. If the energy of the particle is measured, what is the chance of measuring the average of $E_1$ and $E_2$?

2

What else can you say for sure about the result of the measurement?

3

What additional information would allow you to predict for sure that the measurement will produce energy $E_2$, if any?

4

If the energy of a large number of particles, each with the above initial wave function, is measured, what can you say about the results?

5

Suppose the wave function is initially

$$\Psi = {\textstyle\frac{3}{5}} \psi_1 + {\textstyle\frac{4}{5}} \psi_2$$

where $\psi_1$ is an normalized eigenfunction of the kinetic energy operator with eigenvalue $T_1$, and $\psi_2$ an eigenfunction with eigenvalue $T_2$. What are the chances of measuring the kinetic energy to be $T_1$?