PHYSICS 2D
PROF. HIRSCH
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QUIZ 4
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
FEBRUARY 26, 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 4
WINTER QUARTER 2016
FEBRUARY 26, 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
;
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T = e -2αΔx ;
T =e
x1
;
α (x) =
2m[U(x) - E]
h2
Problem 1
An electron moving in a potential U(x) has wavefunction
2
Ψ(x) = Ce − λx
with λ=1A-2. Find the difference in the potential energy of this electron at x1=0.5A and at
x2=0A, i.e. U(x1)-U(x2).
A: 3.81eV; B: 7.62eV; C: 15.24eV; D: 1.91eV; E: not sure (E always counts 0.31 pts)
Problem 2
An electron in a box of length L=9A has equal probability to be at x=L/6 and at x=L/2,.
What is the lowest energy this electron can have?
A: 1.86eV; B: 4.18eV; C: 10.92eV; D: 16.70eV; E: not sure
Problem 3
A particle is described by the wavefunction:
1
1
Ψ(x) = xe− x/2 for x>0, Ψ(x) = xe x/2 for x<0
2
2
Note that Ψ(−x) = −Ψ(x).
If you know only that the particle is at x ≥ 0 , at what x is it most likely to be found?
A: 0; B: 1; C: 2; D: 4; E: not sure
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PHYSICS 2D
PROF. HIRSCH
QUIZ 4
WINTER QUARTER 2016
FEBRUARY 26, 2016
Problem 4
For the particle described by the wavefunction of problem 3, find Δx.
∞
n!
Hint: use ∫ dx x n e − λx = n +1
λ
0
A: 1.14; B: 4.72; C: 2.28; D: 3.46; E: not sure
Problem 5
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Find Δp for an electron in the lowest energy state of a box of length L=19.73A.
(note that pc has units of energy, so p can be expressed in units eV/c)
A: 628eV/c ; B: 314 eV/c; C: 0 eV/c; D: 100 eV/c; E: not sure
Problem 6
A particle with energy E is incident from the left on a potential step of height U located at
x=0. The wavefunction is
Ψ(x) = e ik1 x + 0.1e −ik1 x ; x < 0
Ψ(x) = 1.1e ik2 x
; x>0
Find the value of the potential step U in terms of E
A: U=0.746E; B: 1.08E; C: U=0.194E; D: U=0.331E; E: not sure
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Problem 7
10,000 electrons of energy 2eV are incident on a square barrier of height 3eV and width
5A. Estimate how many electrons are transmitted.
A: 60; B: 20 ; C: 120 ; D:240; E: not sure
Problem 8
∂
?
∂x
2
A: eikx + e 2ikx ; B: e− x ; C: i coskx − sin kx ; D: coskx + sin kx ; E: not sure
Which of the following functions is an eigenfunction of the operator
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