2.E: The Classical Wave Equation (Exercises)

Solutions to select questions can be found online.

2.1A

Find the general solutions to the following differential equations:

1. \(\dfrac{d^{2}y}{dx^{2}} - 4y = 0 \)
2. \(\dfrac{d^{2}y}{dx^{2}} - 3\dfrac{dy}{dx} - 54y = 0\)
3. \(\dfrac{d^{2}y}{dx^{2}} + 9y = 0 \)

2.1B

Find the general solutions to the following differential equations:

1. \(\dfrac{d^{2}y}{dx^{2}} - 16y = 0 \)
2. \(\dfrac{d^{2}y}{dx^{2}} - 6\dfrac{dy}{dx} + 27y = 0\)
3. \(\dfrac{d^{2}y}{dx^{2}} + 100y = 0 \)

2.1C

Find the general solutions to the following differential equations:

1. \(\dfrac{dy}{dx} - 4\sin(x)y = 0 \)
2. \(\dfrac{d^{2}y}{dx^{2}} - 5\dfrac{dy}{dx}+6y = 0\)
3. \(\dfrac{d^{2}y}{dx^{2}} = 0 \)

2.2A

Practice solving these first and second order homogeneous differential equations with given boundary conditions:

1. \(\dfrac{dy}{dx} = ay\) with \(y(0) = 11\)
2. \(\dfrac{d^2y}{dt^2} = ay\) with \(y(0) = 6\) and \(y'(0) = 4\)
3. \(\dfrac{d^2y}{dt^2} + \dfrac{dy}{dt} - 42y = 0\) with \(y(0) = 2\) and \(y'(0) = 0\)

2.3A

Prove that \(x(t)\) = \(\cos(\theta\)) oscillates with a frequency

\[\nu = \dfrac{1}{2\pi}\sqrt{\dfrac{k}{m}} \nonumber\]

Prove that \(x(t)\) = \(\cos(\theta\)) also has a period

\[T = {2\pi}\sqrt{\dfrac{m}{k}} \nonumber\]

where \(k\) is the force constant and \(m\) is mass of the body.

2.3B

Try to show that

\[x(t)=\sin(\omega t)\nonumber \]

oscillates with a frequency

\[\nu = \omega/2\pi\nonumber \]

Explain your reasoning. Can you give another function of x(t) that have the same frequency. 

2.3C

Which two functions oscillate with the same frequency?

1. \(x(t)=\cos( \omega t)\)
2. \(x(t)=\sin (2 \omega t)\)
3. \(x(t)=A\cos( \omega t)+B\sin( \omega t)\)

2.3D

Prove that \(x(t) = \cos(\omega(t))\) oscillates with a frequency

\[\nu = \dfrac{\omega}{2\pi} \nonumber.\]

Prove that \(x(t) = A \cos(\omega(t) + B \sin(\omega(t))\) oscillates with the same frequency:

\[\nu = \dfrac{\omega}{2\pi}. \nonumber\]

2.4

Show that the differential equation:

\[\dfrac{d^2y}{dx^2} + y(x) = 0\nonumber \]

has a solution

\[ y(x)= 2\sin x + \cos x \nonumber \]

2.7

For a classical harmonic oscillator, the displacement is given by

\[ \xi (t)=v_0 \sqrt{\dfrac{m}{k}} \sin \sqrt{\dfrac{k}{m}} t\ \nonumber \]

where \(\xi=x-x_0\). Derive an expression for the velocity as a function of time, and determine the times at which the velocity of the oscillator is zero.

2.11

Verify that

\[Y(x,t) = A \sin \left(\dfrac{2\pi }{\lambda}(x-vt) \right)\nonumber \]

has a frequency \(\nu\) = \(v\)/\(\lambda\) and wavelength \(\lambda\) traveling right with a velocity \(v\).

2.13A

Explain (in words) how to expand the Hamiltonian into two dimensions and use it solve for the energy

2.13B

Given that the Schrödinger equation for a two-dimensional box, with sides \(a\) and \(b\), is

\[\dfrac{∂^2 Ψ}{∂x^2} + \dfrac{∂^2 Ψ}{∂y^2} +\dfrac{(8π^2mE) }{h^2}Ψ(x,y) = 0 \nonumber \]

and it has the boundary conditions of

\(Ψ(0,y)= Ψ (a,y)=0\) and \(Ψ(o,x)= Ψ(x,b)=0\)

for all \(x\) and \(y\) values, show that

\[E_{2,2}=\left(\dfrac{h^2}{2ma^2}\right)+\left(\dfrac{h^2}{2mb^2}\right). \nonumber\]

2.14

Explain, in words, how to expand the Schrödinger Equations into a three-dimensional box

2.18

Solving for the differential equation for a pendulum gives us the following equation,

\[\phi(x)= c_1\cos {\sqrt{\dfrac{g}{L}}} +c_2\sin {\sqrt{\dfrac{g}{L}}} \nonumber \]

Assuming \(c_1=2\), \(c_3= 5\), \(g=7\) and \(L=3\), what is the position of the pendulum initially? Does this make sense in the real world. Why or why not? (We can ignore units for this problem).

2.23

Consider a Particle of mass \(m\) in a one-dimensional box of length \(a\). Its average energy is given by

\[\langle{E}\rangle = \dfrac{1}{2m}\langle p^2\rangle\nonumber \]

Because

\[\langle{p}\rangle\ = 0\nonumber \]

\[\langle p^2\rangle = \sigma^{2}_{p}\nonumber \]

where \(\sigma_p\) can be called the uncertainty in \(p\). Using the Uncertainty Principle, show that the energy must be at least as large as \(\hbar/8ma^2\) because \(\sigma_x\), the uncertainty in \(x\), cannot be larger than \(a\).

2.33

Prove \(y(x, t) = A\cos[2π/λ(x - vt)]\) is a wave traveling to the right with velocity \(v\), wavelength \(λ\), and period \(λ/v\).