16.1: Exercises - Mainly revision
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You will need to consult your notes from Junior Honours Quantum Mechanics and/or one of the many textbooks on Quantum Mechanics.
1. Given the expansion of an arbitrary wavefunction or state vector as a linear superposition of eigenstates of the operator
use the orthonormality properties of the eigenstates to prove that
Work through the proof in both wavefunction and Dirac notations.
The state
Hint: use the expansion
If the expectation value
and give the physical interpretation of this result.
2. The observables
Show that if
(1) a measurement of
(2) a measurement of
(3) another measurement of
Show that if measurement (1) yields any of the values
3. The normalised energy eigenfunction of the ground state of the hydrogen atom
where
(a) Calculate the normalisation constant,
Alternatively, you can use the computer algebra program Maple if you know how to!
(b) Determine the radial distribution function,
(c) Calculate the expectation value of
(d) Calculate the expectation value of the potential energy,
(e) Calculate the uncertainty,
4. At
(a) Show that this wavefunction is an eigenstate of
Hint: express
(b) Sketch the function, e.g. with a contour plot in the x=0 plane.
(c) Can you identify the Hamiltonian for which this is an energy eigenstate ?