Quantum Mechanics: Fundamental Principles

Questions:

(a) In what sense is quantum mechanics fundamentally indeterministic? Give two physical examples of such indeterminacy in real systems.

(b) Describe the role of the Heisenberg uncertainty principle in undermining the Newtonian description of the state of a system.
(c) Outline the role of the wave function in describing the state of an isolated system in one-dimensional wave mechanics. Is the description given by the wavefunction as complete as can be? Discuss whether the indeterminacy of quantum mechanics arises from our lack of knowledge of the time-evolution of the wave function in between measurements (as described by Schrodinger’s equation), or whether it has another origin.
(d) Describe the role of eigenvalue equations in determining which energy values are possible in a bound system. Are there any situations in which quantum mechanics unambiguously predicts the result of an energy measurement with certainty? How does the collapse of the wave function influence the result of an energy measurement taken immediately after another energy measurement?
(e) Explain the roles of the overlap and coefficient rules in predicting the probability of obtaining a given value when an energy measurement is made in a given state.
(f) Explain the role of Born’s rule in predicting the probability of finding a particle within a given interval? How does the collapse of the wave function influence the result of a position measurement taken immediately after another position measurement?
(g) Finally give a brief summary of the types of prediction that are possible in quantum mechanics. Comment on the extent to which these predictions can (in principle) be compared with experiment by, for example, making repeated measurements.

Solution:

(a) Quantum mechanics is irreducibly indeterministic and some things can-not be predicted, even in principle. For instance (i) when a particular radioactive nucleus will decay, remain a matter of pure chance, and (ii) the emission of photons by atoms is also inherently indeterministic.

(b) In Newtonian mechanics, state of a system can be described by position and momentum of its particles, where all of the particles have accurate values of…..

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