ˆ = ih i+1ihi
TRANSCRIPT
86-311: Quantum Me hani s 1 � Moed B (2019/20)
Le turer: Prof. Eli Barkai, TA: Eyal Wala h
General Instru tions
• Allowed materials � the atta hed equation sheets (5 pages) only. No al ulator.
• Answer 3 out of 4 questions. Ea h question sums up to 33 points.
• You have 3.5 hours.
1 Benzene
For a simple model for an ele tron on a benzene ring made of six arbon atoms, one may write the Hamiltonian
H = −t5
∑
i=0
(|i〉 〈i+ 1|+ |i+ 1〉 〈i|)
where |i〉 is the lo alized basis and |0〉 = |6〉.a) The rotation operator is R |i〉 = |i+ 1〉. Show that H and R ommute. (11 points)
b) Show that for the rotation operator the eigen value α−1has eigen state
1√6
∑
αi |i〉. Determine the possiblevalues for α. (11 points)
) Find the energies and eigen states. Present your results in a table. (11 points)
Figure 1: Benzene Ring
1
2 Finite Potential Well
Consider a parti le moving in a �nite potential well V (x) =
{
V0 |x| > a
0 |x| < a.
a) Assume that V0 is large enough to support at least two bound states. Make a rough sket h of the spatial
wave fun tion of the ground state |0〉, and of the ex ited state |1〉. (5 points)b) Whi h of the following matrix elements will be zero? (4 points)
⟨
0∣
∣x2∣
∣ 0⟩
,⟨
0∣
∣x+ x3∣
∣ 1⟩
,⟨
0∣
∣x2∣
∣ 1⟩
, 〈1 |x| 1〉 . ) Assume only |0〉 and |1〉 are relevant, and V0 → ∞. In this energy basis the Hamiltonian (up to an arbitrary
shift) is approximated with
H =~ω
2
(
1 00 −1
)
.
What is ~ω? (5 points)
d) Let A be a time independent operator in this 2× 2 spa e. Derive an equation of motion for 〈A〉. (6 points)
e) For A =
(
0 11 0
)
and B =
(
0 −ii 0
)
write equations of motion for 〈A〉 and 〈B〉. Obtain the solution in terms
of 〈A (0)〉 and 〈B (0)〉. (7 points)
f) At time t = 0 the system is an eigen state of A with eigen value 1, A |ψ (0)〉 = |ψ (0)〉. Solve for |ψ (t)〉 and al ulate 〈A (t)〉. Verify your result in e. (6 points)
3 2D Hydrogen
Using ylindri al oordinates, the S hrödinger equation for the Hydrogen atom in two dimensions is
− ~2
2µ
[
∂2ψ
∂r2+
1
r
∂ψ
∂r+
1
r2∂2ψ
∂ϕ2
]
− Ze2
rψ = Eψ
a) Using ψ (r, ϕ) = R(r)eimϕ
√2π
for m = 0,±1,±2, . . . and dimensionless variables
ε =~2
2µ
E
e4, x =
2µe2
~2r
and R = y(x)√xthe equation for y (x) reads
d2y
dx2− Cm
x2y +
CZ
xy + εy = 0.
Find Cm and CZ . (8 points)
b) Solve the equation in the limit |x| → 0 and in the limit |x| → ∞. Dis uss the boundary onditions. (8 points)
) Using the anzats
y (x) = x|m|+ 12 e−
√−εxf (x)
some tedious work gives (no need to derive this formula)
d2f
dx2+
[
2 |m|+ 1
x− 2
√−ε
]
df
dx+
1
x
[
Z − (2 |m|+ 1)√−ε
]
f = 0.
Using f (x) =∑
k akxk�nd the iteration rule ak+1 = [?] ak.
Analyze the limit k → ∞ of
ak+1
akand explain. (8 points)
d) Find the energy spe trum
En = −Z2e2
2a0
1
(n+ c)2
where a0 = ~2
µe2is the Bohr radius. What is c? (9 points)
2
4 Ladder Operators Transformation
Let η = cosh θa+ sinh θa†, and η† = cosh θa† + sinh θa be linear ombination of the standard Harmoni Os illator
reation and destru tion operators a and a†. Further let η |α〉 = α |α〉 where α is real.
a) Find
[
η, η†]
. (10 points)
b) In the normalized state |α〉, �nd 〈x〉 and 〈p〉. (10 points)
) Obtain ∆x2 and ∆p2. (10 points)
d) Compare your results with the un ertainty prin iple. (3 points)
3
Mathematics
Fourier Transform
de�nition
f(x) =
∫ ∞
−∞
dk√2π
eikxf(k)
f(k) =
∫ ∞
−∞
dx√2π
e−ikxf(x)
properties
F [af(x) + bg(x)] = aF [f(x)] + bF [g(x)] = af(k) + bg(k)
F [f(x+ x0)] = F [f(x)] eikx0 = f(k)eikx0
F[∫
f(x)dx
]=
1
ikF [f(x)] =
1
ikf(k)
F[dn
dxnf(x)
]= (ik)
n F [f(x)] = (ik)nf(k)
F [f(ax+ b)] =1
af
(k
a
)eikb
Delta Function
de�nitionb∑
n=a
anδnm = cm if a < m < b
∫ b
a
dxf(x)δ(x− x0) = f(x0) if a < x0 < b
properties
δ(x) = δ(−x)
δ(ax) =1
|a|δ(x)
δ(g(x)) =∑i
1
|g′(xi)|δ(x−xi) where xi are the roots of g (x)
Delta and Fourier
δ(x− x0) =
∫ ∞
−∞
dk
2πeik(x−x0)
Integrals
Gaussian ∫ ∞
−∞e−ax2
dx =
√π
a
exponent ∫ ∞
0
xne−axdx =n!
an+1
rational functions∫1
a2 + x2dx =
1
aarctan
(xa
)∫
1√a2 − x2
dx = arcsin(xa
)∫
x
a2 + x2dx =
1
2ln(∣∣a2 + x2
∣∣)∫1√
x2 ± a2dx = ln
(∣∣∣x+√x2 ± a2
∣∣∣)logarithm∫
xn ln (x) dx = xn+1
(lnx
n+ 1− 1
(n+ 1)2
)for n 6= −1
Trigonometry
cos2 (α) + sin2 (α) = 1
sin (α± β) = sinα cosβ ± cosα sinβ
cos (α± β) = cosα cosβ ∓ sinα sinβ
sin (2α) = 2 sinα cosα
cos (2α) = 2 cos2 α− 1
2 sin2(α2
)= 1− cosα
2 cos2(α2
)= 1 + cosα
2 cosα cosβ = cos (α− β) + cos (α+ β)
2 sinα sinβ = cos (α− β)− cos (α+ β)
2 sinα cosβ = sin (α+ β) + sin (α− β)
Hyperbolic Functions
coshx =ex + e−x
2, sinhx =
ex − e−x
2, tanhx =
ex − e−x
ex + e−x
cosh2 x− sinh2 x = 1
sinh (x± y) = sinhx cosh y ± coshx sinh y
cosh (x± y) = coshx cosh y ± sinhx sinh y
1
Taylor
exponent and logarithm
ex =
∞∑n=0
xn
n!, ln (1 + x) =
∞∑n=0
(−1)n+1 xn
n!
rational function1
1− x=
∞∑n=0
xn
trigonometric
sinx =
∞∑n=0
(−1)n
(2n+ 1)!x2n+1 , cosx =
∞∑n=0
(−1)n
(2n)!x2n
hyperbolic
sinhx =
∞∑n=0
1
(2n+ 1)!x2n+1 , coshx =
∞∑n=0
1
(2n)!x2n
Spherical Coordinates
coordinates
x = r sin θ cosϕ , y = r sin θ sinϕ , z = r cos θ
r =√x2 + y2 + z2 , θ = arctan
(√x2 + y2
z
), ϕ = arctan
(yx
)gradient (acts on a scalar f)
∇ =∂f
∂rr +
1
r
∂f
∂θθ +
1
r sin θ
∂f
∂ϕϕ
divergence (acts on a vector ~v = vr r + vθ θ + vϕϕ)
∇ =1
r2∂
∂r
(r2vr
)+
1
r sin θ
∂
∂θ(sin θvθ) +
1
r sin θ
∂
∂ϕvϕ
Laplacian (acts on a scalar f)
∇2 =1
r2∂
∂r
(r2
∂f
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂f
∂θ
)+
1
r2 sin2 θ
∂2f
∂ϕ2
2
Physics
Bohr
quantization
L = n~
radii
rn = aBn2 where aB = ~2
4πε0e2
1
m= 0.529A
energies
En = − e2
4πε0
1
2a0
1
n2= −13.6eV
n2
Basics of QM
Schrodinger equation
i~∂
∂t|ψ〉 = Hop|ψ〉
where Hop =pop2m
+ V (xop)
time independent
Hopφn(x) = Enφn(x)
general wave function
ψ(x, t) =
∞∑n=1
Cnφn(x) exp
(−iEn
~t
)
where Cm =
∫φ?m(x)ψ(x, 0)dx
probability
P (x, t) = |ψ(x, t)|2 dx and P (k, t) = |ψ(k, t)|2dk
normalization ∫ ∞
−∞|ψ(x, t)|2dx = 1
Ehrenfest
d
dt〈p〉 = −〈∇V 〉 and
d
dt〈x〉 = 〈p〉
m
Operators
hermiticity ∫ψ?1fopψ2dx =
∫(fopψ1)
?ψ2dx
〈ψ1 |fop|ψ2〉 = 〈fopψ1|ψ2〉
angular momentum
L2op|lm〉 = ~2l(l + 1)|lm〉 , Lz|lm〉 = ~m|lm〉
L+,op|l,m〉 = ~√l(l + 1)−m(m+ 1)|l,m+ 1〉
L−,op|l,m〉 = ~√l(l + 1)−m(m− 1)|l,m− 1〉
average
〈f (t)〉 = 〈ψ (t) |fop|ψ (t)〉 =∫dxψ?(x, t)fopψ(x, t)
uncertainty
∆x∆p ≥ ~2
∆A∆B ≥ 1
2|〈[A,B]〉|
Heisenberg
i~∂ 〈fop〉∂t
= 〈[fop,Hop]〉+ i~⟨∂fop∂t
⟩building operators
fop =∑
|�nal〉〈initial|
Operators in Position Space
position
xop = x
momentum
pop = −i~∇
angular momentum
L2op = − ~2
sin2 θ
[(sin θ
∂
∂θ
)2
+∂2
∂ϕ2
], Lz,op = −i~ ∂
∂ϕ
Commutators
de�nition
[A,B] = AB −BA
properties
[A,A] = 0
[A,B] = −[B,A]
[A+B,C] = [A,C] + [B,C]
[AB,C] = A[B,C] + [A,C]B
[A,BC] = [A,B]C +B[A,C]
1
[AB,CD] = CA[B,D] +A[B,C]D +C[A,D]B + [A,C]BD
known commutators
[rj,op, pk,op] = i~δjk
[Li,op, Lj,op] = i~εijkLk,op ,[L2op, Li,op
]= 0[
aop, a†op
]= 1
Simple Systems
free particle
ψ(x, t) =
∫dk√2πg(k)eikx−iωt
where g(k) =
∫dx√2πe−ikxψ(x, 0) and ω(k) =
~k2
2m
in�nite potential well
φn(x) =
√2
Lsin
(nπLx)
, En =~2
2m
(nπL
)2
rigid rotor
Hop =L2op
2I, |lm〉 , El =
~2l(l + 1)
2I
Scattering
probability current
J (x) =~mIm
[ψ? (x)
dψ (x)
dx
]transmission and re�ection coe�cients
T =
∣∣∣∣JTJI∣∣∣∣ , R =
∣∣∣∣JRJI∣∣∣∣ where R+ T = 1
Harmonic Oscillator
length scales
σx =
√~mω
and σp =√mω~
eigen functions
φn(x) =1√
2nn!√π
1√σxHn
(x
σx
)exp
[− x2
2σ2x
]eigen energies
En = ~ω(n+
1
2
)ladder operators
a†op =xop√2σx
− ipop√2σp
, aop =xop√2σx
+ ipop√2σp
a†op|n〉 =√n+ 1|n+ 1〉 , aop|n〉 =
√n|n− 1〉
Hamiltonian
Hop = ~ω(a†opaop +
1
2
)coherent states
aop|α〉 = α|α〉 , 〈α|a†op = 〈α|α?
|α〉 = e−|α|2/2∞∑
n=0
αn
√n!|n〉
3D Problems with Spherical Symme-
try
Hamiltonian
Hop =p2r,op2m
+L2op
2mr2op+ V (rop)
wave function
ψ (r, θ, ϕ) = Rnl (r)Yml (θ, ϕ)
radial equation: for R (r) = u (r) /r(− ~2
2m
∂2
∂r2+ Veff (r)
)u(r) = Eu(r)
where Veff (r) =~2l (l + 1)
2mr2+ V (r)
Hydrogen
potential
V (r) = − 1
4πε0
e2
r= −Ke2
r
eigen states
Hop|nlm〉 = En|nlm〉 , En = −K2e2
2aB
1
n2
L2op|nlm〉 = ~2l (l + 1) |nlm〉
Lz,op|nlm〉 = ~m|nlm〉
radial functions
Rnl(r) =
√(2n
aB
)3(n− l − 1)!
2n [(n+ l)!]3
×(
2r
naB
)l
L2l+1n+1
(2r
naB
)exp
[− r
naB
]
Lists
Hermite polynomials
H0 (x) = 1 , H1 (x) = 2x
H2 (x) = 4x2 − 2 , H3 (x) = 8x3 − 12x
spherical harmonics
2
l = 0:
Y 00 (θ, ϕ) =
1√4π
l = 1:
Y 11 (θ, ϕ) = −
√3
8πsin θeiϕ
Y 01 (θ, ϕ) =
√3
4πcos θ
Y −11 (θ, ϕ) =
√3
8πsin θe−iϕ
l = 2:
Y 22 (θ, ϕ) =
1
4
√15
2πsin2 θei2ϕ
Y 12 (θ, ϕ) = −1
2
√15
2πsin θ cos θeiϕ
Y 02 (θ, ϕ) =
1
4
√5
π
(3 cos2 θ − 1
)Y −12 (θ, ϕ) =
1
2
√15
2πsin θ cos θe−iϕ
Y −22 (θ, ϕ) =
1
4
√15
2πsin2 θe−i2ϕ
Hydrogen radial functions
R1,0 (r) =2
a3/2B
exp
[− r
aB
]
R2,0 (r) =1√2
1
a3/2B
(1− r
2aB
)exp
[− r
2aB
]
R2,1 (r) =1√24
1
a3/2B
r
aBexp
[− r
2aB
]
3