PHYSICS 2D
PROF. HIRSCH
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QUIZ 5
Formulas:
Time dilation; Length contraction : Δt = γΔt'≡ γ Δt p ;
L = Lp /γ
; c = 3 ×10 8 m /s
Lorentz transformation : x'= γ (x − vt) ; y' = y ; z' = z ; t'= γ (t − vx /c 2 ) ; inverse : v → -v
uy
ux − v
Velocity transformation : ux '=
; uy '=
; inverse : v → -v
2
γ (1− ux v /c 2 )
1− ux v /c
Spacetime interval : (Δs) 2 = (cΔt) 2 - [Δx 2 + Δy 2 + Δz 2 ]
Relativistic Doppler shift : f obs = f source 1+ v /c / 1− v /c
r
r
Momentum : p = γ mu ; Energy : E = γ mc 2 ; Kinetic energy : K = (γ −1)mc 2
Rest energy : E 0 = mc 2
;
Electron : me = 0.511 MeV /c 2
E=
p 2c 2 + m 2c 4
Proton : mp = 938.26 MeV /c 2
Neutron : mn = 939.55 MeV /c 2
Atomic mass unit : 1 u = 931.5 MeV /c 2
; electron volt : 1eV = 1.6 ×10 -19 J
4
Stefan's law : etot = σT , etot = power/unit area ; σ = 5.67 ×10−8 W /m 2K 4
∞
hc
etot = cU /4 , U = energy density = ∫ u( λ,T)dλ ;
Wien's law : λm T =
4.96kB
0
-E/(kB T )
Boltzmann distribution : P(E) = Ce
8π
hc / λ
8πf 2
Planck's law : uλ ( λ,T) = N λ ( λ) × E ( λ,T) = 4 × hc / λkB T
;
N( f ) = 3
λ
e
−1
c
Photons : E = hf = pc ; f = c / λ ; hc = 12,400 eV A ; k B = (1/11,600)eV /K
Photoelectric effect : eVs = K max = hf − φ , φ ≡ work function; Bragg equation : nλ = 2d sin ϑ
Compton scattering : λ'- λ =
h
(1 − cos θ ) ;
mec
h
= 0.0243A
mec
kq q
kq
kq q
Coulomb force : F = 12 2 ; Coulomb energy : U = 1 2 ; Coulomb potential : V =
r
r
rr
r
r r
Force in electric and magnetic fields (Lorentz force): F = qE + qv × B
1
Z2
ke 2 = 14.4 eV A
Rutherford scattering : Δn = C 2
4
Kα sin (φ /2)
1
1
1
1
Hydrogen spectrum :
= R( 2 − 2 )
;
R = 1.097 ×10 7 m−1 =
λmn
m
n
911.3A
2
2
2
2
Z
ke Z
ke
me (ke )
mev 2
ke 2 Z
Bohr atom : E n = −
= −E 0 2 ; E 0 =
=
= 13.6eV ; K =
; U =−
n
2a0
2
2rn
2h 2
r
hf = E i − E f ; rn = r0 n 2 ; r0 =
a0
Z
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WINTER QUARTER 2016
MARCH 11, 2016
de Broglie : λ =
h
E
;f =
p
h
; a0 =
h2
= 0.529A ; L = me vr = nh angular momentum
me ke 2
; ω = 2πf ; k =
2π
;
λ
Wave packets : y(x,t) = ∑ a j cos(k j x − ω j t), or y(x,t) =
E = hω ; p = hk ;
∫ dk a(k) e
i(kx -ω (k )t )
E=
p2
2m
; ΔkΔx ~ 1 ; ΔωΔt ~ 1
j
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group and phase velocity : v g =
dω
ω
; vp =
;
dk
k
Heisenberg : ΔxΔp ~ h ; ΔtΔE ~ h
PHYSICS 2D
PROF. HIRSCH
QUIZ 5
WINTER QUARTER 2016
MARCH 11, 2016
b
Probability: P(x)dx =| Ψ(x) |2 dx
;
P(a ≤ x ≤ b) =
!c = 1973 eVA
∫ dxP(x)
a
E
-i t
h2 ∂ 2Ψ
∂Ψ
+ U(x)Ψ(x,t) = ih
;
Ψ(x,t) = ψ (x)e h
2
2m ∂x
∂t
∞
h 2 ∂ 2ψ
Time − independent Schrodinger equation : +
U(x)
ψ
(x)
=
E
ψ
(x)
;
∫ dx ψ *ψ = 1
2m ∂x 2
-∞
Schrodinger equation : -
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∞ square well : ψ n (x) =
€
π 2h2n 2
2
nπx
sin(
) ; En =
2mL2
L
L
Harmonic oscillator : Ψn (x) = H n (x)e
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−
mω 2
x
2h
;
h2
= 3.81eVA 2 (electron)
2me
1
p2 1
1
; E n = (n + )hω ; E =
+ mω 2 x 2 = mω 2 A 2 ; Δn = ±1
2m 2
2
2
Expectation value of[Q] :< Q >= ∫ ψ * (x)[Q]ψ (x) dx ; Momentum operator : p =
Eigenvalues and eigenfunctions : [Q] Ψ = q Ψ (q is a constant) ; uncertainty :
(k1 − k 2 ) 2
Step potential : reflection coef : R =
, T = 1− R ;
(k1 + k 2 ) 2
k=
h ∂
i ∂x
ΔQ = < Q2 > − < Q > 2
2m
(E − U)
h2
x2
∫
-2 α (x )dx
Tunneling :
ψ (x) ~ e -α x
;
T = e -2αΔx ;
T =e
;
x1
α (x) =
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2m[U(x) - E]
h2
r
r
r
r -i E t
h2 2
∂Ψ
∇ Ψ + U( r )Ψ( r ,t) = ih
;
Ψ( r ,t) = ψ ( r )e h
2m
∂t
2 2
π h n12 n 22 n 32
3D square well : Ψ(x,y,z) = Ψ1 (x)Ψ2 (y)Ψ3 (z) ; E =
( +
+ )
2m L12 L22 L23
Spherically symmetric potential: Ψn,l,m (r,θ, φ ) = Rnl (r)Ylm (θ, φ ) ; Ylm (θ, φ ) = Plm (θ )e im φ
Schrodinger equation in 3D : -
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l
l
l
l
l
r r r
h ∂
; [L2 ]Ylm l = l(l +1)h 2Ylm l ; [L z ]Ylm l = ml hYlm l
Angular momentum : L = r × p ; [Lz ] =
i ∂φ
ke 2 Z 2
Radial probability density : P(r) = r 2 | Rnl (r) |2 ;
Energy : E n = −
2a0 n 2
1 Z
Ground state of hydrogen and hydrogen - like ions : Ψ1,0,0 = 1/ 2 ( ) 3 / 2 e−Zr / a 0
π
a0
→
→
€ Orbital magnetic moment : µ = −e L ; µz = −µB ml ; µB = eh = 5.79 ×10−5 eV /T
2me
2me
r
1
−e r
Spin 1/2 : s = , | S |= s(s + 1)h ; Sz = msh ; ms = ±1/2 ;
µs =
gS
2
2me
g=2
r
−e r
r r
r
µ=
( L + gS ) ; Energy in mag. field : U = −µ⋅ B
Orbital + spin mag moment :
2me
r r
r r
Two particles : Ψ(r1, r2 ) = + /− Ψ( r2 , r1 ) ; symmetric/antisymmetric
Screening in multielectron atoms : Z → Z eff , 1 < Z eff < Z
Orbital ordering: 1s<2s<2p<3s<3p<4s<3d<4p<5s<4d<5p<6s<4f<5d<6p<7s<6d~5f
PHYSICS 2D
PROF. HIRSCH
QUIZ 5
WINTER QUARTER 2016
MARCH 11, 2016
Problem 1
An electron in a three-dimensional cubic box makes a transition from an excited state to
the ground state and emits a photon of wavelength 5000A. What is the smallest possible
side length of this box?
A: 0.28A; B: 2.36A; C: 4.12A; D: 6.74A; E: not sure (E always counts 0.31 pts)
Problem 2
An electron is in a two-dimensional rectangular box with one side twice as long as the
other side. The lowest energy level has energy 3eV. What is the energy of the lowest
energy level that is doubly degenerate?
A: 6eV; B: 8eVeV; C: 12eV; D: 16eV; E: not sure
Problem 3
The wavefunction for an electron in a hydrogen-like ion with nuclear charge Ze is
ψ (r, θ , φ ) = Cr 2 e−2r/a0 sin θ cosθ eiφ . The values of the quantum numbers n, l, ml are
A: 3, 2, 1; B: 4, 3, 1; C: 4, 2, 1; D: 3, 1, 1; E: not sure
Problem 4
For the wavefunction of problem 3, Z is:
A: 2; B: 4; C: 6; D: 8; E: not sure
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Problem 5
The radial wavefunction for an electron in a hydrogen-like ion is
R(r) = Cre −r / a 0
The average radius, i.e. < r > , is
A: a0 ; B: 2a0; C: 1.5a0; D: 2.5a0; E: not sure
∞
n!
Hint: use ∫ dx x n e − λx = n +1
λ
€
0
Problem 6
1
For the radial wavefunction of problem 5, the average < > is
r
€
A: 1/a0 ; B: 1/(2a0); C: 1/(1.5a0); D: 1/(2.5a0); E: not sure
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Problem 7
A gas of hydrogen atoms absorbs photons of€wavelength approximately 1216A, When a
magnetic field of magnitude 8T is turned on, the spectral line splits into several closely
spaced lines. The difference in wavelength between the highest and lowest line is
approximately:
A: 0.02A; B: 0.1A; C: 0.5A ; D:3A; E: not sure
(Assume normal Zeeman effect)
Problem 8
In an atom of atomic number Z=50, the most loosely bound electron has orbital quantum
number
A: l = 0 ; B: l = 1; C: l = 2 ; D: l = 3 ; E: not sure
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