PHYSICS 4E
PROF. HIRSCH
QUIZ 3
SPRING QUARTER 2010
MAY 7
Formulas and constants:
hc = 12,400 eV A ; k B = 1/11,600 eV/K ; ke 2 = 14.4eVA ; me c 2 = 0.511×10 6 eV ; m p /me = 1836
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Relativistic energy - momentum relation E = m2 c 4 + p 2 c2
;
c = 3 × 108 m / s
r
r
r r
Photons: E = hf ; p = E /c ; f = c / λ
Lorentz force: F = qE + qv × B
1
Photoelectric effect : eV0 = ( mv 2 ) max = hf − φ , φ ≡ work function
2
∞
∞
2
dIn
1 π
1
x3
π4
€Integrals : In ≡ ∫ x n e− λx dx ;
€ ; I1 =
= −In +2 ; I0 =
; ∫ x dx =
dλ
2 λ
2λ
15
0
0 e −1
_
_
8π
hc
1
Planck's law : u( λ) = n( λ ) E ( λ) ; n( λ) = 4
;
E ( λ) =
hc / λkB T
λ
λ e
−1
−E / kB T
Energy in a mode/oscillator : E f = nhf ; probability P(E) ∝ e
∞
Stefan's law : R = σT 4 ; σ = 5.67 ×10−8 W /m 2K 4 ; R = cU /4 , U =
0
Wien's displacement law : λm T = hc /4.96k B
h
h
Compton scattering :
λ'- λ =
(1− cos θ )
;
λc ≡
= 0.0243A
mec
mec
kq Q
1
Rutherford scattering: b = α 2 cot(θ /2) ;
ΔN ∝ 4
mα v
sin (θ /2)
kq1q2
kq
Electrostatics : F = 2 (force) ; U = q0 V (potential energy) ; V =
(potential)
r
r
1
1
1
1
Hydrogen spectrum
:
= R( 2 − 2 ) € ;
R = 1.097 ×10 7 m−1 =
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λ
m
n
911.3A
Bohr atom : rn = r0 n 2 ; r0 =
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∫ u(λ)dλ
a0
Z2
h2
ke 2
; E n = −E 0 2 ; a0 =
=
0.529A
;
E
=
= 13.6eV ; L = mvr = nh
0
Z
n
mke 2
2a0
1
ke 2 Z
ke 2 Z
v2
E k = mv 2 ; E p = −
; E = Ek + E p ; F = 2 = m
; hf = hc/λ = E n − E m
2
r
r
r
mM
Reduced mass : µ =
; X - ray spectra : f 1/ 2 = An (Z − b) ; K : b = 1, L : b = 7.4
m+M
h
E
2π
p2
de Broglie : λ = ; f =
; ω = 2πf ; k =
; E = hω ; p = hk ; E =
; hc = 1973 eV A
p
h
λ
2m
group and phase velocity : v g =
dω
dk
;
vp =
ω
k
;
Heisenberg
: ΔxΔp ~ h
ΔtΔE ~ h
;
Ψ(x,t) =| Ψ(x,t) | e iθ (x,t ) ;
P(x,t) dx =| Ψ(x,t) |2 dx = probability
E
-i t
h2 ∂ 2Ψ
∂Ψ
h
Schrodinger equation : +
V(x)Ψ(x,t)
=
ih
;
Ψ(x,t)
=
ψ
(x)e
2m ∂x 2
∂t
∞
h 2 ∂ 2ψ
Time − independent Schrodinger equation : + V(x)ψ (x) = Eψ (x) ; ∫ dx ψ *ψ = 1
2
2m ∂x
-∞
Wave function
∞ square well : ψ n (x) =
2
nπx
π 2h2n 2
sin(
) ; En =
;
L
L
2mL2
x op = x , pop =
h ∂
; < A >=
i ∂x
Eigenvalues and eigenfunctions : Aop Ψ = a Ψ (a is a constant) ; uncertainty :
Harmonic oscillator : Ψn (x) = Cn H n (x)e
mω 2
−
x
2h
2
∞
*
∫ dxψ A
ψ
op
-∞
ΔA = < A 2 > − < A > 2
1
p
1
1
; E n = (n + )hω ; E =
+ mω 2 x 2 = mω 2 A 2 ; Δn = ±1
2
2m 2
2
PHYSICS 4E
PROF. HIRSCH
QUIZ 3
(k1 − k2 ) 2
Step potential : R =
,
(k1 + k2 ) 2
T = 1− R
SPRING QUARTER 2010
MAY 7
;
k=
2m
(E − V )
h2
b
∫
-2 α (x )dx
Tunneling :
ψ (x) ~ e -αx
;
T ~ e -2αΔx ;
T~e
a
;
α (x) =
€
2m[V (x) - E]
h2
π 2 h 2 n12 n 22 n 32
( +
+ )
2m L12 L22 L23
Justify all your answers to all problems. Write clearly.
3D square well : Ψ(x,y,z) = Ψ1 (x)Ψ2 (y)Ψ3 (z) ; E =
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Problem 1 (10 pts)
An electron is in the ground state of an infinite one-dimensional well of width 8A.
(a) Estimate the probability that the electron is in a region within 1A of the center of the well,
and compare with the classical answer.
(b) If you were to calculate this probability exactly, would it be larger or smaller than the
answer you gave in (a)? Justify.
(c) Give the uncertainty in the momentum of this electron, Δp, in units eV/c.
(d) For extra credit, do after you finish the rest of the quiz: calculate the probability in (a)
exactly.
Problem 2 (10 pts)
The ground state energy of an electron in a harmonic oscillator potential is 3eV.
(a) What is the wavelength of photons emitted and absorbed by this system, in A?
(b) What is the classical amplitude of oscillation (=classical turning point) when the electron
is in the ground state, in Angstrom?
Use h 2 /2me = 3.81eV A 2
(c) How much more likely is it to find the electron in the ground state at position x=0 than
at a classical turning point?
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Problem 3 (10 pts)
left
V0
2a
electron
is here
L=10A
2V0
right
a
An electron is in the one-dimensional well of length L=10A shown above, bounded by the
two barriers of height V0 and 2V0 and widths 2a and a respectively.
(a) For what energy range of this electron is it more likely to escape the well through the
right barrier than through the left barrier? Give your answer in terms of V0 .
(b) Assuming you can approximate the energy levels of the well by the infinite well energy
levels, for what quantum number n is the electron equally likely to escape through the right
as through the left barrier, for V0 =127eV?
(c) If you now took into account that the energy levels in a finite well are not exactly the
same as for the infinite well, would the particle in the level n found in (b) be more likely to
escape through the right or the left barrier? Justify your answer. You can answer this (with
justification) even if you didn't find the answer to (b).
Justify all your answers to all problems. Write clearly.