Exercise 5.1.1 Show that Eq. (5.1.9) may be rewritten as an integral over and a sum over the index as
Exercise 4.2.1 Consider the following operators on a Hilbert space :
(1) What are the possible values one can obtain if is measured?
(2) Take the state in which . In this state what are , and ?
(3) Find the normalized eigenstates and the eigenvalues of in the basis.
(4) If the particle is in the state with , and is measured, what are the possible outcomes and their probabilities? (5) Consider the state
in the basis. If is measured in this state and a result is obtained, what is the state after the measurement? How probable was this result? If is measured, what are the outcomes and respective probabilities?
(6) A particle is in a state for which the probabilities are , , and . Convince yourself that the most general, normalized state with this property is
It was stated earlier on that if is a normalized state then the state is a physically equivalent normalized state. Does this mean that the factors multiplying the eigenstates are irrelevant? [Calculate for example .]
Exercise 2.1.1 Consider the following system, called a harmonic oscillator. The block has a mass and lies on a frictionless surface. The spring has a force constant . Write the Lagrangian and get the equation of motion.