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 the state , the average of any physical observable for this system in 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 x 2 . L1.P1 Lecture 1 Page 1
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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.
L1.P1
Lecture 1 Page 1
Solution
L1.P2
Lecture 1 Page 2
L1.P3
Lecture 1 Page 3
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.
L1.P4
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?
Lecture 1 Page 4
Solution
(3) The spectrum is doubly degenerate, for a given n there are two eigenfunctions