Homework 1: due Friday, Jan 10

Homework is due Friday by 2 PM in the Physics Dept mailroom across the hall from room ISB 231. There will be box to put your HW in from about noon to 2 PM. (If you come earlier or don't see the box, then you can put it in my mailbox.) 

Please do not hesitate post questions or comments here. I encourage you to question anything that seems incorrect or unclear to you. Your questions and comments are strongly encouraged and appreciated. I am also interested to get feedback on whether you find some of these problems too difficult or too easy. Feel free to respond to other students questions and comments. Peer-to-peer dialogue can be very valuable.

This assignment focuses on math-related things which will be relevant to something we will cover in the near future. Feel free to use Wolfram-alpha for integrals, help with graphing, etc.. Graphs should be hand sketched --not too big (or too small). (Two relationships that did not make it onto this assignment are: exp(i*theta)= cos(theta) + i sin(theta), and (e^a)*(e^b) = e^(a+b).)

Waves:
Imagine a string attached to fixed posts at either end. The string's simplest motions, are standing waves, called normal modes, in which the string moves at one particular frequency.  For problem 1 you are asked to sketch the string displacement profile at a time of maximum displacement (e.g., t=0) as a function of x.
1. a) For a wave on a string with fixed ends at x=-L/2 and L/2, sketch the lowest frequency mode.
b) Write a mathematical expression for this lowest frequency mode (at a time of maximum overall displacement).
c) Sketch the next lowest frequency mode of a wave on a string (with fixed ends at x=-L/2 and L/2).
d) Write a mathematical expression for y(x) for this mode (at maximum displacement).

Graphing: (Things to notice: overall shape, node location, asymptotic behavior. Please label and/or put scales on axes.)
2. (These are relevant to square well states.)
a) sketch a graph of cos(pi x/L) from x = -L/2 to L/2.
b) sketch a graph of sin(kx) from x = -pi/k to pi/k.
c) sketch a graph of cos(.7 pi x/L) from x = -L/2 to L/2. What is its value at x=L/2?
d) sketch a graph of B e^{-(x-L/2)/a} from x= L/2 to (L/2 + 4a). what is its value at x=L/2? At x=(L/2)+2a ?



3. (These are relevant to harmonic oscillator states.)
a) sketch a graph of e^{-x^2/a^2} (from about x = -2a to 2a)
b) sketch a graph of (x/a) e^{-x^2/a^2} (from about x = -2a to 2a)
c) sketch a graph of (x/a)^2 e^{-x^2/a^2} (from about x = -2a to 2a)
d) sketch a graph of (x/a)^2 e^{-(x-2a)^2/a^2} (from about x = -2a to 6a)

4.
a) Sketch a graph of  |x|.
b) Sketch a graph of e^(-|x|/a). (|x| means "absolute value of x")
c) Sketch graphs of:  |x-2a|,  and of e(-|x-2a|/a).

Calculation:
5. (These are relevant to expectation value calculations, normalization and orthogonality.)
a) Integrate e^(-x^2/a^2) from x = -infinity to infinity.
b) Integrate (x/a) e^(-x^2/a^2) from x = -infinity to infinity.
c) Integrate (x/a)^2 e^(-x^2/a^2) from x = -infinity to infinity.
d) What are the units of the integrand and of the integral for 5 a)? (x has units of length).
e) Integrate (1-2x^2/a^2)e^(-x^2/a^2) from x = -infinity to infinity.
f) calculate exp{-i*w*t) at t=0, t=pi/(2w), and t=pi/w. Write exp{-i*w*t) in terms of cos and sin functions.
g) calculate the 2nd derivative of e^(-x^2/2a^2) with respect to x. (extra credit:)What are the units of each term in your result? (relevant to quantum kinetic energy)

Units for the quantum world:  (These will be used a lot.)
6. Using eV and nm (rather than Joules, meters, or some other choice) :
a) What is mc^2 ?
b) What is hbar*c ?
c) Find the value of is e^2/(4*pi*epsilon_0) in units of eV-nm. (e is the charge of an electron; pi = 3.14…; epsilon_0 is the permittivity of free space.)
d) Find the value of  hbar^2/m in eV-nm^2  (where m is the mass of an electron).

Spherical Coordinates: (7 and 8 are relevant to the hydrogen atom.)
7. a) what is z in spherical coordinates? (write an expression for z in terms of r, theta and phi.)
b) what is x in spherical coordinates?
c) what is y in spherical coordinates?

8. (These are relevant to the hydrogen atom.)
a) In spherical coordinates, calculate the integral of 1 over all theta and phi, and then from r = zero to a.
b) Calculate the integral over all (3D) space of: (1/pi*a^3))*r^2*exp{-2r/a}.
c) what are the units of your result from 8b?

Linear Algebra:  (9-11 are relevant to understanding state choices in the excited states of the H atom (and any other quantum degeneracy). You could skip these problems if you are pressed for time. The trade-off is that then your understanding of hydrogen states may not be as deep as if you understand these basic linear algebra concepts.)

9. In the context of linear algebra:
a) what is a basis?
b) given a set of vectors, what does it mean to say they are "linearly independent"?
c) given a set of vectors, what does it mean to say they "span"?
d) given a set of vectors, what does it mean to say they are "orthogonal" (or mutually orthogonal) ?
e) what is an orthonormal basis?
f) are these concepts familiar?

10.
Consider the set of vectors {{1,1,1},{1,0,-1},{2,1,0}} .
a) Are they linearly independent?
b) Are they (mutually) orthogonal?
c) Do they span 3-D space?

11.
Consider the set of vectors {{1,1,1},{1,0,-1},{-1,2,-1}} .
a) Are they linearly independent?
b) Are they (mutually) orthogonal?
c) Do they span 3-D space?

Extra credit:  Does one of the integrals from problem 5 illustrate an orthogonality relationship? What particular orthogonality relationship is it?