Download Modern Physics (PHY 251) Lecture 26 Spin Multi

Modern Physics (PHY 251)
Lecture 26
Joanna Kiryluk
Spring Semester Lectures
Department of Physics and Astronomy, Stony Brook University
§ Spin
§ Multi-electron atoms, Table of Elements
Textbook: 8.1, 8.5
8.3 optional/extra reading
Lecture25 Atomic Hydrogen and Hydrogen-Like Ions
Z =1
Z >1
Hydrogen atom:
1 proton + 1electron
Hydrogen Like ions:
Z protons + A-Z neutrons + 1electron
e.g. He+ , Li2+
kZe
V r =−
r
()
2
Coulomb potential
Spherical coordinates
2
The Hydrogen-like atom constitutes the central force problem
Lecture25

 iωt
m
iω t Electron is in a state described by
Ψ ( r, t ) = ψ (r )e = Rnl (r)Yl (θ , φ )e
three quantum numbers n, l and ml
l
Use tables (next slide)
ke 2 " Z 2 %
En = −
$ 2'
2a0 # n &
n = 1, 2, 3,....
o
2
a0 =
= 0.529 A
2
me ke
Important!
Ground state:
ke 2 2
E1 = −
Z = − (13.6eV ) Z 2
2a0
Bohr radius
n
principal quantum number
l = 0,1, 2,..., ( n −1) orbital quantum number
−l ≤ ml ≤ l
magnetic quantum number

 iωt
m
Lecture25
Wave function: Ψ nlm ( r, t ) = ψ nlm (r )e = Rnl (r)Yl (θ , φ )eiωt
l
l
l
Electron is in a state described by three quantum numbers n, l and ml
Lecture25
(assumption:
no spin)
Examples:
n
principal quantum number
l = 0,1, 2,..., ( n −1) orbital quantum number
−l ≤ ml ≤ l
magnetic quantum number
1.
If n=1, then
•
l=0, and ml =0
n=1:
2.
If n=2, then
• l=0, and ml =0
• l=1, and ml = -1,0,1
n=2: 4 possible states
3.
If n=3, then
n=3: 9 possible states
• l=0, and ml =0
• l=1, and ml = -1,0,1
• l=2, and ml = -2, -1,0,1,2
1 possible state
(ground state)
and so on (n=1,2, …. Infinity)
In general, for a particle there are n2 possible states. Every state
is characterized by a set of 3 quantum numbers (n,l and ml)
The Hydrogen-like atom constitutes the central force problem
Lecture25

 iωt
ml
iω t Electron is in a state described by
Ψ ( r, t ) = ψ (r )e = Rnl (r)Yl (θ , φ )e
three quantum numbers n, l and m
Use tables (next slide)
ke 2 " Z 2 %
En = −
$ 2'
2a0 # n &
n = 1, 2, 3,....
o
2
a0 =
= 0.529 A
2
me ke
Important!
Ground state:
ke 2 2
E1 = −
Z = − (13.6eV ) Z 2
2a0
Bohr radius
n
principal quantum number
l = 0,1, 2,..., ( n −1) orbital quantum number
−l ≤ ml ≤ l
magnetic quantum number
L2 Ψ nlm = l(l +1)!2 Ψ nlm %%%%%%%%l = 0,1,2,...
Lz Ψ nlm = ml !Ψ nlm %%%%%%%%%%%ml = 0,±1,±2,....,±l
For hydrogen-like atoms the quantum numbers n, l(l+1) and ml
are associated with the “sharp” observables E, L2 and Lz
l
Physics interpretation of l and ml quantum numbers
Lecture25
m
Θ(θ )Φ(φ ) = Yl l (θ , φ ) − spherical harmonics
l, ml − quantum numbers for angular
momentum only
L2 Ψ nlm = l(l +1)!2 Ψ nlm %%%%%%%%l = 0,1,2,... n-1
Lz Ψ nlm = ml !Ψ nlm %%%%%%%%%%%ml = 0,±1,±2,....,±l
More in QM course
7
Lecture25 Probabilities (electron in hydrogen-like atoms)
State n,l,ml :
§ Wave function:

 iωt
Ψ nlml ( r, t ) = ψ nlml (r )e = Rnl (r)Yl ml (θ , φ )eiωt
2

 2
 iωt 2
ml
§ Probability density: Pnlml ( r, t ) = Ψ nlml ( r, t ) = ψ nlml (r )e
= Rnl (r)Yl (θ , φ )
2 2

ml
Pnlml ( r, t ) dV = Rnl (r)Yl (θ , φ ) rsin
drd
θ d
φ
θ


∫ Pnlml (r, t ) dV =
Normalization condition:
∫
dV
∫

Pnlml ( r, t ) dV =
total
volume
ml
2
Rnl (r)Yl (θ , φ ) r 2 sin θ dr dθ d φ
∞ π 2π
∫∫∫
ml
2
Rnl (r)Yl (θ , φ ) r 2 sin θ dr dθ d φ = 1
0 0 0
Radial probability density (see also next slide)
Def: P( r) = r2 |Rnl (r )|2
infinity
Average r =⎰ rPnl(r ) dr
0
r position where an electron is most likely to be found:
Use the following condition: dPnl(r )/dr =0
Lecture25
§ Ground state:
Example:
Ground state of the hydrogen-like atom
n = 1, l = 0, ml = 0
§ Wave function of the electron in the ground state:

ψ100 (r ) = R10 (r)Y00 (θ , φ )
3/2
3/2
!Z$
1
where : R10 (r) = # & 2e−Zr/a0 and Y00 (θ , φ ) =
2 π
" a0 %
(from tables)

1 !Z$
−Zr/a
ψ100 (r ) =
# & 2e 0
2 π " a0 %
§ Radial probability density of the electron in the ground state:

 2 2
P100 ( r ) dV = ψ 100 (r ) r sin θ drdθ dφ
3
∫∫
1 ! Z $ −2 Zr/a0 2
r dr sin θ dθ d φ =
# & 4e
4π " a0 %
3
1 ! Z $ −2 Zr/a0 2
4Z 3 −2 Zr/a0 2
=
r dr4π = 3 e
r dr
# & 4e
4π " a0 %
a
0

 Radial probability density
P100 (r )
for the electron in ground state
Lecture25
Example:
Ground state (n=1, l=0, ml=0) of the hydrogen-like atom
P( r) = r2 |R10 (r )|2
10
Lecture25
Spectroscopic notation (chemistry):
§ States with the same quantum number n form a shell
§ Shells are identified by letters K,L,M,…. which designate the states for
which n=1,2,3,….
§ States which have the same value of both n and l form a subshell
§ The letters s,p,d,f,… are used to designate the states for which l=0,1,2,3,…
n
l = 0,1, 2,..., ( n −1)
−l ≤ ml ≤ l
n, l, ml − integers
11
Spin
Atomic Structure
W. Pauli
N. Bohr
spinning top
The discovery of the electron spin, S.A. Goudsmit:
https://www.lorentz.leidenuniv.nl/history/spin/goudsmit.html
Spin – Internal Angular Momentum
§ 1925: Uhlenbeck and Goudsmit proposed that each electron rotates with
angular momentum  2 (~10-34 Js) and carries µ B = e 2 m
G.E.Uhlenbeck
H.Kramers
Paul Ehrenfest (1925):
“This is a good idea.
Your idea may be wrong,
but since both of you are
so young and without any
reputation, you would
not loose anything making
a stupid mistake.”
S.A.Goudsmit
…. spin (internal angular momentum)
is a fundamental property of particles …
Spin: purely quantum mechanical phenomenon,
does not exist in classical mechanics !
13
Result: no Nobel Prize for the spin discovery
https://www.library.ethz.ch/exhibit/pauli/elektronenspin_e.html
https://www.youtube.com/watch?v=jDxUaBYINeQ
https://www.youtube.com/watch?v=cd2Ua9dKEl8
https://www.youtube.com/watch?v=3k5IWlVdMbo
Adding spin to the hydrogen wavefunction:
Quantum Mechanics course (beyond the scope of this lecture).
Here we’ll discuss total angular momentum of an electron to
explain the atomic structure of elements.
s − spin quantum number
ms = −s, −s +1,.., s −1, s
Defined similarly as for
For a given s: (2s+1) possible values of ms the orbital momentum
quantum number l and ml
S 2 = s(s +1) 2 − spin angular momentum
Sz = ms 
§ Spin quantum number is intrinsic property of a particle
§ Spin angular momentum is quantized
s − spin quantum number
ms = −s, −s +1,.., s −1, s
Defined similarly as for
For a given s: (2s+1) possible values of ms the orbital momentum
S 2 = s(s +1) 2 − spin angular momentum quantum number l and ml
Sz = ms 
§ Spin quantum number is intrinsic property of a particle
§ Spin angular momentum is quantized
Electron (fermion):
s=
1
2
ms = ±
electron’s spin quantum number (dimensionless)
1
2
(2*1/2+1)=2 possible values of ms
3 2
electron’s spin angular momentum squared (in units of  2 )
S = 
4
1 1
Sz = − ,  electron’s z-component of spin angular momentum (in units of  )
2 2
2
s − spin quantum number
ms = −s, −s +1,.., s −1, s
Defined similarly as for
For a given s: (2s+1) possible values of ms the orbital momentum
S 2 = s(s +1) 2 − spin angular momentum quantum number l and ml
Sz = ms 
§ Spin quantum number is intrinsic property of a particle
§ Spin angular momentum is quantized
Electron (fermion):
s=
1
2
ms = ±
1
2
3 2
S = 
4
1 1
Sz = − , 
2 2
2
Electron (fermion):
s=
1
2
1
ms = ±
2
3
S = 2
4
1 1
Sz = − , 
2 2
2
Photon (boson):
s =1
ms = −1, 0,1
S 2Extra
= 3 2
material
Sz = −, 0, 
20
§ The coupling of spin and orbital momentum implies that
neither orbital angular momentum nor spin angular
momentum is conserved separately.
§ Total angular momentum is conserved.
  
J = L+S
§ Vector addition
§ Quantized
Quantum rules for adding angular momenta:
l−s ≤ j ≤l+s
- j ≤ mj ≤ j
J 2 = j( j +1) 2 − total angular momentum
Jz = m j
21
!
Total angular momentum:
SUMMARY
  
J = L+S
§ Vector addition
§ Quantized
Quantum rules for adding angular momenta:
l−s ≤ j ≤l+s
- j ≤ mj ≤ j
J 2 = j( j +1) 2 − total angular momentum
Jz = m j
Example: electron: s=1/2 (always, property of electron)
s − spin quantum number
l-orbital momentum quantum number
l= 0,1,….,n
ms = −s, −s +1,.., s −1, s
ml = -l, -l+1, ….. , l-1, l
2
2
S = s(s +1) − spin angular momentum
Sz = ms 
L2=l(l+1)ℏ2
Lz=mlℏ
Electron (spin=1/2 particle)
l −1 2 ≤ j ≤ l +1 2
Application
m j = − j,...., j
J 2 = j( j +1) 2 − total angular momentum
Jz = m j
For example, if l=1:
l = 1 ⇒ j = 1 2, 3 2
j = 1 2, m j = −1 2, 1 2
J 2 = 3 2 4, J z = − 2,  2
j = 3 2, m j = −3 2, −1 2, 1 2, 3 2
J 2 = 15 2 4, J z = −3 2, −  2,  2, 3 2
25
Energy Levels: Spectroscopic Notation
nl j
n-principle quantum number,
l –orbital momentum quantum number
j-total angular momentum quantum number
Examples:
n = 1, l = 0 :1s1/2
n = 2, l = 0 : 2s1/2
n = 2, l = 1: 2 p3/2 , 2 p1/2
26
The Pauli Exclusion Principle
No two electrons can be in the same quantum state
simultaneously.
An electron quantum state (in a hydrogen-like atoms) is a
state described by n, j, m j quantum numbers:
1
1
l− ≤ j ≤l+
2
2
for each j, mj can take 2j+1 values:
− j ≤ mj ≤ j
Example: if for two electrons n, l, and ml are the same, ms
must be different such that the electrons have opposite spins
27
The Periodic Table of Elements
Atomic number Z = number of protons in the nucleus
Hydrogen Z=1
Helium Z=2
An electrically neutral atom contains bound electrons equal in
28
number to the protons Z in the nucleus.
The Periodic Table of Elements
Consequence of the Pauli exclusion principle:
the electron shell structure of atoms.
For example, the electron shell structure explains the variety of
chemical elements.
l = 0,1,..., n −1
l−
1
1
≤ j ≤l+
2
2
e.g. Ar (Argone)
Z=18
Shell structure:
2
2
6
2
1s , 2s , 2 p , 3s , 3p
26
− j ≤ mj ≤ j
Number of electrons
Each shell can hold at maximum 2n2 electrons:
n=1: 2 electrons (l=0, j=1/2), n=2: 8 (2+6) electrons (l=0 j=1/2; l=1 j=1/2,3/2),
n=3: 18 (2+6+10) (l=0 j=1/2; l=1 j=1/2,3/2 ; l=2 j=3/2,5/2) electrons
Extra
§
Hund's rule:
Every orbital in a subshell is singly occupied with one
electron before any one orbital is doubly occupied
and
§
all electrons in singly occupied orbitals have the same spin.
30
Extra
Lithium (Z=3)
Beryllium (Z=4)
Boron (Z=5)
Carbon (Z=6)
Nitrogen (Z=7)
Oxygen (Z=8)
Fluorine (Z=9)
Neon (Z=10)
Ne
31
Extra
Ionization Energy
Noble gasses,
Closed shells
One electron outside a closed shell
(core electrons screening effect)
32
Final exam:
December 19, 2016 (Monday) 5:30pm-8pm
Place: Javits 102
Material: Lectures 1-25,
Practice problems, Recitations, Homeworks
Thank you!
Concluding remarks,
this course and in general:
1. fixed mindset:
2 possible mindsets
• Value looking smart over learning
• Effort – bad
• Failures define you
Versus YOU:
2. growth mindset:
• Value learning and effort
• See mistakes and setbacks
as tools for learning
Effect of growth mindset
• Take on more challenges
• Bounce back from setbacks
• Higher achievement
Scientific evidence: Perseverance and practice can change your
brain by developing new neural pathways (biology)