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- Notes
- Analysis I (undergraduate), MIT Open Courseware, Math Deptartment
- Quantum Theory I (graduate), MIT Open Courseware, Physics Department
- Statistical Mechanics (graduate), MIT Open Courseware, Chemistry Department
- Billingham J. and King A.C., Wave Motion, CUP (2000)

This is a stub page for posting homework problems. Focused work on the problems in a book can take place on a dedicated page.

Many problem sets can be found here:

- MIT OpenCourseWare

(original)

**Problem 1.1**. Prove that there is no rational number whose square is 12.

**Problem 1.2**. Let S be a non-empty subset of the real numbers, bounded above. Show that if u = sup S, then for every natural number n, the number u - 1/n is not an upper bound of S, but the number u + 1/n is an upper bound of S.

**Problem 1.3**. Show that if A and B are bounded subsets of R, then A UNION B is a bounded subset of R. Show that sup(A UNION B) = max{sup A, sub B}.

**Problem 1.5**. Prove that no order can be defined in the complex field that turns it into an ordered field. (Hint: -1 is a square.)

**Problem 1.1**. A skew-Hermitian operator A is an operator satisfying A-dagger = -A.

(a) Prove that A can have at most one real eigenvalue (which may be degenerate).

(b) Prove that the commutator of two Hermitian operators is skew-Hermitian.

**Problem 1.2**. Show that if H and H are both Hermitian operators with positive eigenvalues, then Tr HK >= 0, and that equality implies that HK = 0.

**Original**: here

**Problem 1.1**. Consider a random walk on a linear three site model. p and q are the probabilities of moving right and left, respectively. (p+q=1)

TODO: insert diagram, or use textual description.

1) Write the transition matrix Q.

2) Find the stationary distribution and show that is satisfies detailed balance.

3) For the special case of p = q = 1/2, compute the probability at n=3 after s steps.

4) Repeat the calculation in 3) for p != q.

**Exercise 1.1** Find the dispersion relation for the propagation of plane harmonic waves in the systems:

*a) The free-space Shroedinger equation*

i*du/dt + d2u/dt2 = 0*

*b) The linearised 5th order KdV equation*

dphi/dt + a*dphi/dx = b* d3phi/dx3 = g*d5phi/dx5*

In each case, determine both the phase and group velocities. For each equation, determine phi(x,t) if phi(x,0’)= d(x).

Use the method of stationary phase to find an approximation to this solution for t » 1 with v = x/t = O(1), v constant.

**Exercise 1.2** Consider the equation

psi_xxtt = -g*a*psi_xx

where g and a are positive constants.

This equation arises as a model for internal waves in a fluid.

Calculate the dispersion relation, and hence determine the solution when:

psi(x,0) | abs x > 1 = 0 | abs x <= 0 = 1

psi_t(x,0) = 0.

**Exercise 1.3** Show that:

E = integral 0 1 (0.5*m*(eta_xx^2) + (B*d4eta/dx4)= 0*

is a conserved quantity if n satisfies the *linearised plate bending equation*:

m*d2eta/dt2 + B*d4eta/dx4 = 0

subject to the boundary conditions:

n_xx= n_t = 0 at x = 1,l

x = (1,l) where l is length.

E is the bending energy of the plate. By considering to waves of nearly equal wavenumber, show that the bending energy propagates at the group velocity.

category: problem solving