Lecture#1 Review Thestateofasystematanyinstantof ...msafrono/425-2009/Lecture 1.pdf · Lecture#1 Review Postulatesofquantummechanics (1-3) Postulate 1 Thestateofasystematanyinstantof

Lecture #1

Review

Postulates of quantum mechanics (1-3)

Postulate 1

The state of a system at any instant of time may be represented by a wave function which is continuous and differentiable. Specifically, if a system is in thestate , the average of any physical observable for this systemin time is

Only normalizable wave functions represent physical states. The set of all square-integrable functions, on a specified interval,

constitutes a Hilbert space. Wave functions live in Hilbert space.

Exercise 1

Problem 1.5

Consider the wave function

where A, λ, and ω are positive real constants.

(a) Normalize

(b) Determine the expectation values of x and x2.

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Solution

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Postulate 2

To any self-consistently and well-defined observable Q , such as linear momentum,energy, angular momentum, or a number of particles, there correspond an operator such that measurement of Q yields values (call these measured values q) whichare eigenvalues of Q. That is, the values q are those for which the equation

has a solution . The function is called the eigenfunction of corresponding

to the eigenvalue q.

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Exercise 2

Consider the operator

angle in polar coordinates, and the functions are subject to

(1)Is hermitian?(2) Find its eigenvalues and eigenfunctions.(3) What is the spectrum of Q? Is the spectrum degenerate?

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Solution

(3) The spectrum is doubly degenerate, for a given n there are two eigenfunctions

except special case n=0, which is not degenerate.

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