Homework (3) for Physics 316, Modern Physics I (Hostaetter/Drasco/Thibault) Due Date: Friday, 02/18/04 - 9:05 in 132 Rockefeller Hall Exercise 1: Replace the relativistic treatment of de Broglie which lead to p = k and E = p2 with E = (pc)2 + (mc2 )
Homework (4) for Physics 316, Modern Physics I (Hostaetter/Drasco/Thibault) Due Date: Friday, 02/18/05 - 9:05 in 132 Rockefeller Hall Exercise 1: Show that whenever a solution (x, t) of the time-dependent Schrdinger equation sepao rates into a produc
Homework for Physics 316, Modern Physics I (Hostaetter/Drasco/Thibault) Due Date: Friday, 03/04/05 - 9:55 in 132 Rockefeller Hall Exercise 1: The oscillation amplitude x of a damped classical harmonic oscillator with the force d Cx b dt x and the ma
1 Physics 316 Cornell University Solution for homework 5 Spring 2005 while for the second, we have A = x0 B = x0 + x0 .
I. NOTATION
2
(2.7) (2.8)
Throughout these exercises we will occasionally make use of a notation known as Diracs bra-ket notat
1 Physics 316 Cornell University Solution for homework 6 Spring 2005 where we have used the known Gaussian integral
2
dx eAx =
I. EXERCISE 1
2
. A
(1.8)
To prove this relation let I = Compute the position uncertainty x in the ground state 0
Homework (7) for Physics 316, Modern Physics I (Hostaetter/Drasco/Thibault) Due Date: Friday, 03/18/05 - 9:55 in 132 Rockefeller Hall Exercise 1: How can a device be built that outputs horizontally polarized light of an intensity that is independent
1 Physics 316 Cornell University Solution for homework 8 Spring 2005
2 In the last case, we need to know what is the direction of polarization. To do so, we only need to replace the R/L analyser with a linear polarizer, say, in the x direction. We a
Homework (9) for Physics 316, Modern Physics I (Hostaetter/Drasco/Thibault) Due Date: Friday, 04/08/05 - 9:55 in 132 Rockefeller Hall Exercise 1: In Electrodynamics the charge density (x, t) and the current density j(x, t) satisfy a continuity equati
Homework (10) for Physics 316, Modern Physics I (Hostaetter/Drasco/Thibault) Due Date: Friday, 04/15/05 - 9:55 in 132 Rockefeller Hall Exercise 1: A particle in an innite square well extending between x = 0 and x = L has the wave function (x, t) = A(
1 Physics 316 Cornell University Solution for homework 10 Spring 2005
II. EXERCISE 2 A. I. EXERCISE 1 A.
2
A particle in an innite square well extending between x = 0 and x = L has the wave function (x, t) = A 2 sin x i E1 t e h + sin L 2x L ei
E2
1 Physics 316 Cornell University Solution for homework 11 Spring 2005
u(r) r.
2 the actual wavefunction being given by (r) = We can solve (2.1) in the two domains r < R and r > R independantly and then make sure that u(r) and its rst derivative are
Homework for Physics 316, Modern Physics I (Hostaetter/Drasco/Thibault) Due Date: Monday, 05/02/05 - 9:55 in 132 Rockefeller Hall Exercise 1: The atomic weight of iron is 55.85, and solid iron has a density of 7.8g/cm 3 . When a strong magnetic eld i
Homework for Physics 316, Modern Physics I (Hostaetter/Drasco/Thibault) Due Date: Friday, 05/06/03 - 9:55 in 132 Rockefeller Hall Exercise 1: Compute the radial part Rnl (r) of the Hydrogen wave function for n < 3 for all possible l. Show by direct i
7
Derivation of black body radiation
01/26/2005
The first quantum property discussed was that of light by Max Plank (1900). By proposing light quanta he derived the function f ( ) T Derivation: 1. Represent the black body as a black body box. 2.
1 Physics 316 Cornell University Solution for homework 12 Spring 2005
IV. EXERCISE 4 A. I. EXERCISE 1 A.
2
Consider a hydrogen atom described by a time-dependent wave function (ignoring spin) that has the following structure at t = 0:
3
The atomic
1 Physics 316 Cornell University Solution for homework 13 Spring 2005 For instance, for R20 , we nd: 1 = |A|2 a0 = |A|2
0
2
I.
EXERCISE 1 A.
1 ex x x2 2 ex x2 dx 4! ) 4
0
2
dx
0
ex x3 dx + 1 A= . 2a0
1 4
ex x4 dx
0
Compute the radi
1
0RGHUQ 3K\VLFV
Photons and the quantum of light Simple model of the atom The wave properties of particles Wave-particle duality and bound states Solutions of Schrdingers equation in one dimension Applications of Schrdingers equation Photons and qu
12
Average energy in each mode for light quanta
In statistical mechanics one learns that at temperature T, H the probability P for a state to have energy E is: P e kT Since P is normalized to 1,
01/28/2005
, k = 1.381 10 23
J K
P=e
H kT
1
15 01/31/2005
Further evidence for photons
The photoelectric effect: 1) The maximum electron energy depends only on Q and increases linearly with Q: Emax = hQ - W (Millikan 1916) 2) There is a minimum Q independent of intensity. It is the Work funct
149
Expectation values for px
Review of one dimension: < < x2 < x1 x3
04/20/2005 04/02/2003
n +1 ( xn +1 ) = n ( xn + ) 2 2
< x
n = An e i ( kn
xn t )
n +1 = An +1e i ( kn+1 xn+1 t )
An+1 = An e
ik n ( xn + ) ik n+1 ( xn+1 ) 2 2
146
Normalization and probability density
Probability to find a particle in the volume element Probability to find the particle somewhere is one:
04/18/2005
& 2 3& is given by | ( x , t ) | d x & 2 3& | ( x, t ) | d x = 1
& dx
3
Examples:
141
04/15/2005 Spherically symmetric potentials and wave functions & + y + z ) + V ( x ) = E 3D Schrdinger equation: 2 m ( x & & & & dx = (dr er + rd e + r sin d e ) = dr r + d + d & & & & & & & & ek ek , er = e , er = sin e ,
134
Expectation values for px
Review of one dimension: < < x2 < x1 x3
04/13/2005 04/02/2003
n +1 ( xn +1 ) = n ( xn + ) 2 2
< x
n = An e i ( kn
xn t )
n +1 = An +1e i ( kn+1 xn+1 t )
An+1 = An e
ik n ( xn + ) ik n+1 ( xn+1 ) 2 2
123
Potential step down
k12+kk2 eik1x + k12kk2 e ik1x for x < 0 1 ( x ) = C 1 ik x e2 for x 0
04/08/2005
( x, t ) =
(
many
A ( x)e i t
6FDWWHULQJ RFFXUV HYHQ ZKHQ WKH SRWHQWLDO VWHS JRHV GRZQ ZKLFK FODVVLFDOO\ ZRXOG QRW OHDG WR D UHIOH
20
The Rutherford-Bohr Atom
n = 2 cRH ( 1 n1 ) 4
2
The Balmer lines of Hydrogen
Visible range: 430-750 THz 700-400 nm 1.8 3.1 eV photons
02/02/2005
Rutherfords scattering experiment (1909) lead to the insight that an atom has a positively c
22
Stimulated emission for balck-body radiation Einsteins explanation from 1917 for the energy density u ( ) in a black body box.
The light in a black body is emitted by electrons that change their energy level:
02/04/2005
N j B ji u ( ) N i j =
24
2. Wave properties of particles
02/07/2005
De Broglie (1924): Light waves regularly have quantization conditions (e.g. standing waves) and also show particle character in photons. Electrons have particle character, but also show quantization co
39
The stationary Schrdinger equation
i x = k
Conclusion: whenever k needs to be computed, one can use
2 x2
02/16/2005
i x
= i ( x k ) + ik x
A wave function changes significantly over a wavelength. Whenever the potential does not change
27
George Thomsons experiment
02/09/2005
In a powdered, microcrystalline substance there is always some crystal which has the correct angle for constructive interference 2d cos = n Diffraction pattern Each ring corresponds to one type of crystal