Physics  401:  Quantum  Mechanics  I

Chapter  4

Are  you  here  today?

A. YesB. NoC. After  than  midterm?

3-­D  Schroedinger  Equation

The  ground  state  energy  of  the  particle  in

a  3D  box  is    

What  is  the  energy  of  the  1st excited  state?  

• A) 4ε B) 5ε C) 6ε D) 8ε E) 9ε

12 +12 +12( ) π

22

2ma2 = 3ε .

83

In  the  3D  infinite  square  well,what  is  the  degeneracy  of  the  energy

corresponding   to  the  state(nx ,  ny ,  nz )  =  (1,  2,  3)?  

A) 1 B) 3 C) 4 D) 6 E) 9

84

Spherical  coordinates

In  Cartesian  coordinates,  the  volume  element  is  dx  dy  dz.  In  spherical  coordinates,  the  volume  element  is

A) r2sinθ cosφ dr dθ dφB) sinθ cosφ dr dθ dφC) r2cosθ sinφ dr dθ dφD) r sinθ cosφ dr dθ dφE) r2sinθ dr dθ dφ

99

A  planet  is  in  elliptical  orbit  about  the  sun.  

The  torque,                                    on  the  planet  about  the  sun  is:

A) Zero alwaysB) Non-zero alwaysC) Zero at some points, non-zero at

others.95

€

τ = r × F

P A

The  magnitude  of  the  angular  momentum  of  the  planet  about  the  sun                                    is:

A) Greatest at the perihelion point, PB) Greatest at the aphelion point, AC) Constant everywhere in the orbit

96

€

L = r × p

P A

Are  you  here  today?

A. YesB. NoC. WWHD?

In  Cartesian  coordinates,  thenormalization  condition  is

In  spherical  coordinates,  the  normalization  integral  has  limits  of  integration:

E) None of these

€

dx−∞

∞

∫ dy−∞

∞

∫ dz−∞

∞

∫ Ψ2

=1.

100

€

A) dr0

+∞

∫ dθ0

2π

∫ dϕ0

π

∫ …

€

B) dr−∞

+∞

∫ dθ0

2π

∫ dϕ0

π

∫ …

€

C) dr0

+∞

∫ dθ0

2π

∫ dϕ0

2π

∫ …

€

D) dr−∞

+∞

∫ dθ0

π

∫ dϕ0

π

∫ …

A  and  B  are  positive  constants.  r  is  radial  distance  (0  ≤  r  <  ∞).    Using  the  whiteboards:

Sketch                                    and    

What  does  the  graph                                                      look  like?

rA−

111

€

Br2

€

y(r) =Br2−Ar

Are  you  here  today?

A. YesB. NoC. WWHD?

Hydrogen  Energy  Levels

Consider  He+  (1  e-­ around  a  nucleus,    Q=  2e).  If  you  look  at  “Balmer  lines” (e-­ falling  from  higher  n  down  to  n=2)  what  part  of  the  spectrum  do  you  expect  the  emitted  radiation  to  fall  in?

€

En =E1

n2 , with E1 = −(ke2)2me /22

A)VisibleB)  IRC)  UVD)  It’s  complicated,  not  obvious  at  all.  

Recall,  for  hydrogen:

The  spectrum  of  "Perkonium"  has  only  3  emission  lines.

114

wavelength(nm)200 300 400 500 600

5  eV 3  eV 2  eV

Which  energy  level  structure  is  consistent  with  the  spectrum?

E(eV)

–2–3

–5

–2

–4–5

–7 –7

–5

–10

–5

–7–8

(A) (B) (C) (D)

As  indicated  in  the  figure,  the  n  =  2,  l =  0  state  and  the  n  =  2,  l  =  1  state  happen  to  have  the  

same  energy  (given  by  E2  =    E1/22 ).Do  these  states  have  the  same  radial  

wavefunction  R(r)  ?

x

Veff

l =  0

l =  1 l =  2

n  =  1

n  =  2n  =  3

4

A)  Yes B)  No

Radial  probability:  ground  state  of  hydrogen

120

Consider  an  electron  in  the  ground  state  of  an  H-­atom.  The  wavefunction  is  

0(r) Aexp( r / a )ψ = −Where  is  the  electron  more  likely  to  be  found?

A)Within  dr  of  the  origin  (r  =  0)

B)  Within  dr  of  a  distance  r  =  a0 from  the  origin?

A

B

x

y

r  =  a0

Are  you  here  today?

A. YesB. NoC. Eigenfunction,  eigenvalue,  eigenfunction,  eigenvalue…          what’s  the  difference  again?

Hydrogen  atom  orbitals

We  are  solving  the  equation

What,  then,  is  the  full  3-­D  wave  function  for  hydrogen  atom  stationary  states?A) u(r,θ,φ) B)  u(r)Ylm(θ,φ)

C)  ru(r)Ylm(θ,φ)              D)  r2u(r)Ylm(θ,φ)

E)  None  of  these  

−2

2md 2udr2

+−ke2

r+2l(l +1)2mr2

"

#$

%

&'u = Eu

True  (A)  or  False  (B)

Any  arbitrary  stationary  state  of  an  electron  bound  in  the  H-­atom  potential  can  always  be  written  as  

with  suitable  choice  of  n, l, and  m.

ψ n,l ,m (r,θ ,φ) = Rnl (r)Ylm (θ ,φ),

Suppose  at  t=0,  

Is Ψ (r,t) given very simply byΨ (r,t =0) e-iEt/ħ?

A)Yes, that’s the simple resultB) No, it’s more complicated (a superposition of two states with different t dependence => “sloshing”)

€

Ψ(r,t = 0) =12(ψ210 +ψ200)

Suppose  at  t=0,  

€

Ψ(r,t = 0) =12(ψ 200 +ψ 300)

Is Ψ (r,t) given very simply byΨ (r,t =0) e-iEt/ħ?

A)Yes, that’s the simple resultB) No, it’s more complicated (a superposition of two states with different t dependence => “sloshing”)

“Allowed”  1-­photon  transitions

How  many  of  the  following  transitions  to  the  2p  in  an  H-­atom  will  result  in  emission  of  a  photon  ?  

s p d f

n  =  1

2

3

4

E

A) all  of  them:  11        B)  None  of  them:  0              C)  8

D)  9 E)  6

0

1

2

3

4

5

6

7

8

9

10

11

12

13

0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100+

Fre

qu

en

cy

Score

P402 W16 Exam Distribution

Midterm Final

Midterm  results

Average = 70

Are  you  here  today?

A. YesB. NoC. WWHD?

The  angular  momentum  operator  is

with  e.g.  

Is              Hermitian?    (Hint:  Is  Lz Hermitian?)

A) Yes B) NoC) Only Lz is  (Lx and  Ly are  not)  D) Lz is  not    (but  Lx and  Ly are)  E) Are you joking here? Can I do this as a

clicker question?

€

ˆ L x = ˆ y p z − ˆ z p y, ..., ˆ L z = ˆ x p y − ˆ y p x

€

ˆ L = ˆ r × ˆ p = ( ˆ L x, ˆ L y, ˆ L z )

€

ˆ L

Is  the  commutator,

zero  or  non-­zero?

A) ZeroB) Non-zero

€

ˆ x , ˆ p y[ ]

86

The  commutator,

zero  or  non-­zero?

A) ZeroB) Non-zeroC) Sometimes zero, sometimes non-zero

€

ˆ y p z, ˆ x p z[ ]

97

The  commutator,

zero  or  non-­zero?

A) ZeroB) Non-zeroC) Sometimes zero, sometimes non-zero

€

Lz2, ˆ L z[ ]

98

Are  you  here  today?

A. YesB. NoC. How is the cat doing anyhow?

Angular  momentum  raising  and  lowering  operators

Are  you  here  today?

A. YesB. NoC. What about the dog?

Postulates  of  Quantum  Mechanics(1)  The  state  of  a  particle  is  completely  represented  by  a  normalized  vector  in  Hilbert  space,  which  we  call  |ψ>

(2)  All  physical  observables,  Q,  are  associated  with  Hermitian  operators  Q,  and  the  expectation  value  of  Q  in  some  state  |ψ>   is  <ψ|Q|ψ>(3)  A  measurement  of  Q  on  a  particle  in  state  |ψ>  is  certain  to  return  a  particular  value,  λ  ,  iff ("if and  only  if")  Q|ψ> =  λ|ψ>,  (i.e.  if  and  only  if  |ψ>  is  already  an  eigenvector  of  Q,  with  eigenvalue  λ)

(3a)  If  you  measure  Q  in  any state  |ψ>,  you  are  certain  to  obtain  one  of  the  eigenvalues  of  Q.  The  probability  of  measuring  some  eigenvalue  λ is  given  by  |<uλ|ψ>|2,  where  |uλ>  is  defined  to  be  the  eigenvector  of  Q with  eigenvalue  λ

(3b)  After  a  measurement  gives  you  the  value  λ,  the  system  will  collapse  into  the  state  |uλ>  

(4)  The  time  evolution  of  the  state  |ψ>  is  given  by  Schrodinger's  equation:

i∂ψ∂t

= H ψ

= 2

Are  you  here  today?

A. YesB. NoC. It depends on what kind of spin

you put on it.

SJP QM 3220 3D 1

Page H-32 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008

(ThispagefromQMnotesofProf.RogerTobin,PhysicsDept,TuftsU.)

Stern-Gerlach Experiment (W. Gerlach & O. Stern, Z. Physik 9, 349-252 (1922).

€

F = ∇ µ • B ( ) = (

µ • ∇ ) B (in current free regions), or here,

F = ˆ z (µ

z

∂Bz

∂z)(thisisalittle

crude‐seeGriffithsExample4.4forabettertreatment,butthisgivesthemainidea)

Deflectionofatomsinz‐directionisproportionaltoz‐componentofmagnetic

momentµz,whichinturnisproportionaltoLz.Thefactthattherearetwobeamsis

proofthatl=s=½.Thetwobeamscorrespondtom=+1/2andm=–1/2.Ifl=1,

thentherewouldbethreebeams,correspondingtom=–1,0,1.Theseparationof

thebeamsisadirectmeasureofµz,whichprovidesproofthat

Theextrafactorof2intheexpressionforthemagneticmomentoftheelectronis

oftencalledthe"g‐factor"andthemagneticmomentisoftenwrittenas

.Asmentionedbefore,thiscannotbededucedfromnon‐relativistic

QM;itisknownfromexperimentandisinserted"byhand"intothetheory.

Schematic  of  Stern-­Gerlach  experiment

Stern-­Gerlach  experiments  #1  and  2

#1

#2

State  preparer State  analyzer

Stern-­Gerlach  experiment  #3

Stern-­Gerlach  experiment  #4