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