{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

3063_ProblemSet7

# 3063_ProblemSet7 - PHY3063 Spring 2007 Problem Set 7 PHY...

This preview shows pages 1–2. Sign up to view the full content.

PHY3063 Spring 2007 Problem Set 7 Department of Physics Page 1 of 3 PHY 3063 Problem Set #7 Due Thursday April 5 (in class) (Total Points = 110, Late homework = 50%) Reading: Finish reading Tipler & Llewellyn Chapter 5 and 6. Problem 1 (25 points): . (a) (2 points) Show that the sum of two hermitian operators is hermitian (b) (2 points) Suppose that H op is a hermitian operator, and α is a complex number. Under what condition (on α ) is α H op hermitian? (c) (2 points) When is the product of two hermitian operators hermitian? (d) (2 points) If dx d O op = , what is op O ? (e) (2 points) Show that = op op op op A B B A ) ( . (f) (2 points) Prove that [AB,C] = A[B,C] + [A,C]B, where A, B, and C are operators. (g) (2 points) Show that dx df i x f p op x h = )] ( , ) [( , for any function f(x). (h) (2 points) Show that the anti-hermitian operator, I op , has at most one real eigenvalue (Note: anti-hermitian means that op op I I = ). (i) (2 points) If A op is an hermitian operator, show that 0 2 >≥ < op A . (j) (2 points) The parity operator, P op , is defined by P op Ψ (x,t) = Ψ (-x,t). Prove that the parity operator is hermitian and show that 1 2 = op P , where 1 is the identity operator. Compute the eigenvalues of the parity operator. (k) (2 points) Idempotent operators have the property that op op P P = 2 . Determine the eigenvalues of the idempotent operator P op and characterize its eigenvectors. (l) (3 points) Unitary operators have the property that 1 = = op op op op U U U U , where 1 is the identity operator. Show that the eigenvalues of a unitary operator have modulus 1.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 3

3063_ProblemSet7 - PHY3063 Spring 2007 Problem Set 7 PHY...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online