## Lecture (xx minutes)

Discussion of the concept of superposition of wave functions and the implications for measurement, projection, probability of measuring a particular eigenvalue, normalization, expectation values. The most tractable example is the infinite square well potential energy.

This is an opportunity to pose many questions of the type: If the wave function is of the type (and use specific values) $\psi \left( x \right) = {c_1}{\varphi _1}\left( x \right) + {c_2}{\varphi _2}\left( x \right) + {c_3}{\varphi _3}\left( x \right) + …$, then

• is the wave function normalized? why or why not?
• what is the probability that a measurement of energy yields a particular energy, say ${E_3}$?
• what is the projection of the wave function onto a particular state, say ${\phi_2}$?
• if the energy is measured of a series of identically prepared states, what is the average value of the energy?
• if the momentum is measured of a series of identically prepared states, what is the average value of the momentum?
• if the energy is measured of a series of identically prepared states, what is the average value of the position?
• what is the probability of finding an electron between two particular values of $x$?

An important point is that any of these calculations can be carried out in wave function form or in bra-ket form. The key is to recognize that if energy is the variable in question, the explicit form of the wave function is not needed - only the fact that the state is an eigenfunction, and that eigenstates are orthonormal. Calculating average values of $x$ requires explicit use of the wave function.

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