{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# homework1 - tions 3 Consider the equation of a damped...

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

Homework 1 PHZ 5156 Due Thursday, August 31 1. Consider a particle of mass m trapped in a two-dimensional box with infinitely high walls at x = 0, x = a , y = 0, and y = b . Inside the box, the potential is zero and the Hamiltonian is given by ˆ H = - 2 2 2 m = - 2 2 m 2 ∂x 2 + 2 ∂y 2 (1) The infinitely high potential walls that trap the particle result in the boundary conditions for the wave function ψ ( x = 0 , y ) = 0, ψ ( x = a, y ) = 0, ψ ( x, y = 0) = 0, and ψ ( x, y = b ) = 0. a) Use the method of separation of variables to write the time-independent Schrodinger equation ˆ ( x, y ) = ( x, y ) (2) as two ordinary differential equations. b) Determine the solutions to the two equations obtained in part a). c) What are the allowed energies E ? 2. The torsion of a bar is described by the fourth-order equation, d 4 θ dx 4 + τθ = 0 (3) Show how this can be expressed as a system of coupled first-order differential equa-

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.

Unformatted text preview: tions. 3. Consider the equation of a damped, driven simple harmonic oscillator, m d 2 y dt 2 + γ dy dt + ky = Fcos ( ωt ) (4) a) Find the particular solution y ( t ) to this equation. Write your answer in the form y ( t ) = | A | cos ( ωt + δ ), and determine expressions for | A | and the phase angle δ . To keep your work simple, use the deﬁnition for the natural frequency of the oscillator ω = q k m . 1 b) Determine the condition for resonance. In other words, ﬁnd an expression for the driving frequency ω which results in a maximum amplitude | A | for the response. 4. Write down ﬁrst-order ﬁnite diﬀerence approximations for each of the follow-ing: a) df dx b) d 2 f dx 2 c) d 3 f dx 3 d) ∂ ∂t ∂f ∂x 2...
View Full Document

{[ snackBarMessage ]}