Download Representation Theory: Applications in Quantum Mechanics

Representation Theory:
Tackling Problems in Quantum Mechanics
KRISTINA CHANG
PROJECT ADVISOR: RAMIN NAIMI
SENIOR COMPREHENSIVE IN MATHEMATICS, SPRING 2016
Meet the wavefunction
wavefunction 𝜓 𝒓, 𝑡
1) Wavefunctions are solutions to…
t-dependent Schrodinger Eq:
𝑯 ߰(r,t)=𝑖ℏ
𝜕
𝜓
𝜕𝑡
where 𝑯 is the Hamiltonian Operator:
𝐫, t
H=T+V
2) Stationary states (special 𝝍’s) are solutions to…
t-independent Schrodinger Eq:
where E is a constant
𝑯 𝜑 𝑟 = 𝐸𝜑 𝑟
Symmetry?
GOAL: Solve for 𝜑 in cases were H has intrinsic symmetry using Representation Theory.
KRISTINA CHANG
REPRESENTATION THEORY IN QUANTUM MECHANICS
SPRING 2016
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Roadmap
Roadmap
KRISTINA CHANG
REPRESENTATION THEORY IN QUANTUM MECHANICS
SPRING 2016
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Representing a mathematical entity A with a
mathematical entity B.
What is a
representation?
Group
A
(H)
Matrices
B
Why is this useful?
•Translate problems in QM involving H (symmetrygroup) into a linear algebra
4
What is a representation?
Def. A representation of a group G is a group homomorphism φ: G → GL(n,C).
Group homomorphism:
φ(g1 g2) = φ(g1) φ(g2)
g1
g2
g3
φ
𝒂 𝒃
𝒄 𝒅
𝒆 𝒇
𝒈 𝒉
G
KRISTINA CHANG
…
𝒊
𝒌
𝒋
𝒍
GL(2,C)
REPRESENTATION THEORY IN QUANTUM MECHANICS
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What groups can we represent?
Symmetry group:
The symmetry of an object can be classified by a set of “actions” – symmetry operations – which
preserve its position, shape, and orientation.
y
2
CC44CC4
Composition of rotations forms a closed group.
4
1
2
x
4
KRISTINA CHANG
Notation:
◦ 𝐂𝐧
rotate by 𝝅/n
◦ ( Cn )m = 𝐂𝐧𝐦 rotate by (𝝅/n), m times
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REPRESENTATION THEORY IN QUANTUM MECHANICS
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What groups can we represent?
Consider…
< Cn > = {Cn0 , Cn1 , … , Cnn−1 }
y
“Pure Rotation Groups”  Cyclic!!!
1
2
How can we represent < Cn > with matrices?
x
Rotations are linear transformations!
Rθ →
𝑐𝑜𝑠 𝜃
𝑠𝑖𝑛 𝜃
4
− 𝑠𝑖𝑛 𝜃
𝑐𝑜𝑠 𝜃
3
Intuitive representation: Let φ: <C4> → GL(2,C) be given by
𝐶4𝑚 →
KRISTINA CHANG
𝑐𝑜𝑠 𝜃𝑚
𝑠𝑖𝑛 𝜃𝑚
− 𝑠𝑖𝑛 𝜃𝑚
𝑐𝑜𝑠 𝜃𝑚
where 𝜃𝑚 = m 𝜋/4
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At this point, we have
1) seen a formal definition of a representation
Irreducible
representations
2) explored one intuitive way to construct a
representation
We will soon introduce reducibility (a property of a
representation).
But first… we’ll need a few important concepts:
 New notation
Ways to generate new rep’s from old ones
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Sizes
and
Sums
A note about
notation:
Suppose G = { g1, g2, g3, …, gn }, and let φ be a representation of G.
We will often refer to φ by the image set of φ:
{ Γ∝ | ∝ =1, 2, …, n }
where Γi = φ(gi) for all i.
For convenience, we will abbreviate “ { Γ∝ } ”
Remember:
{ Γ∝ } =
𝒂 𝒃
,
𝒄 𝒅
KRISTINA CHANG
𝒆 𝒇
,
𝒈 𝒉
…,
REPRESENTATION THEORY IN QUANTUM MECHANICS
𝒊 𝒋
𝒌 𝒍
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Let a representation φ of G be given by φ: G → GL(n,C).
Sizes
Sums
Def 1. The and
dimensionality
of the representation φ is n.
Suppose
∝: G → GL(n,C) and
{ [∝ (𝑔1 )],
[∝ (𝑔2 )], …, [∝ (𝑔 𝐺 )] }
Def.
β: G → GL(m,C) are representations of G.
{ [β(𝑔1 )],
[β(𝑔2 )], …, [β(𝑔 𝐺 )] }
The direct sum of ∝ and β is the homomorphism ∝⊕β: G → GL(n+m,C) given by (∝⊕β)(g) =
[∝ (𝑔)]
00
∝ g 𝑔∈𝐺 G
[∝ (𝑔1 )] (∝⊕β)(g)
0
[∝
𝑔2 ]
=
for
all
,
, …,
0
[β(𝑔)]
0
[β(𝑔1 )]
0
[β(𝑔2 )]
0
KRISTINA CHANG
REPRESENTATION THEORY IN QUANTUM MECHANICS
0
β 𝑔𝐺
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Reducibility
Th. Suppose φ: G → GL(n,C) is a representation of G.
If φ = ∝⊕β, for some functions ∝ and β, then ∝ and β are themselves
representations of G.
Def. us
If there
matrix P such that for all g in G,
Leads
to ideaisofsome
reducibility:
-1
φ(g)sum
= P of∝(g)⊕β(g)
P
Rep φ can be decomposed into a direct
lower-dimensional
rep’s ∝ and β
for some nontrivial functions ∝ and β, then φ is a reducible representation of G.
Else, φ is an irreducible representation of G.
KRISTINA CHANG
REPRESENTATION THEORY IN QUANTUM MECHANICS
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Practice using def. of irreducibility:
Reducibility
All 1-dimensional representations of a group are irreducible.
Th.
“Proof.”
There are no representations with a dimension less than 1.
 1-dimensional representation can never be decomposed as a direct sum
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Consequences of Irreducibility
Groups have a complete set of irreducible representations.
(basis for reducible representations)
Linear algebra applied to the def. of irreducibility…
Great Orthogonality Theorem - stringent criteria on irreducible rep’s
 Allows us to derive a complete set of all irreducible representations for most common
symmetry groups…
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REPRESENTATION THEORY IN QUANTUM MECHANICS
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Irreducible representations for < 𝐶𝑁 >
where 𝜔 = 𝑒 2𝜋𝑖/𝑁
Complete Table of Irreducible Rep’s
Matrices in
the image
set of the
irr. rep.
KRISTINA CHANG
𝐸
𝐶𝑁
Γ1
Γ2
1
1
1
𝐶𝑁 2
1
𝜔
𝜔2
…
…
⋮
Γ𝑁
⋮
1
𝜔𝑁−1
𝜔𝑁−2
…
REPRESENTATION THEORY IN QUANTUM MECHANICS
…
𝐶𝑁 𝑁−1
1
Elements of group
𝜔𝑁−1
2
(𝑁−1)
𝜔
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So far…
Quantum
Mechanics
• We showed how a symmetry group can be
represented using matrices
• Presented a complete set of irreducible
representations for cyclic groups
What’s this got to do with Quantum Mechanics?
Recall:
We want to utilize symmetries in H to make solving for
𝜑 ′ 𝑠 easier. (Hint: H belongs to a symmetry group!)
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Group of the Hamiltonian
The Symmetry of the Hamiltonian
Suppose R is an operation under which H is invariant:
R-1 H R = H
Eq. 6.2
If R satisfies Eq. 6.2, we say that R is a symmetry operation of the Hamiltonian H.
The Group of the Hamiltonian
Th.
The set of all symmetry operations {𝑹𝜶 } of a Hamiltonian forms a group.
This means we can represent {𝑹𝜶 } !
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REPRESENTATION THEORY IN QUANTUM MECHANICS
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Time-independent SE:
H𝜑=E𝜑
The Schrödinger Equations
• Eigenvalue equation!
eigenvalue E, eigenfunction 𝝋 𝒓
Properties: (parallels to Linear Algebra!)
1) Distinct E’s have orthogonal 𝝋’s
2) E may have multiple orthogonal 𝝋’s.
If E does not have multiple orthogonal 𝝋’s, then E is non-degenerate.
If E has exactly n distinct eigenfunctions, E is n-degenerate.
KRISTINA CHANG
REPRESENTATION THEORY IN QUANTUM MECHANICS
SPRING 2016
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We are now almost ready to prove “The Bridging
Theorem:”
“The Bridging
Theorem”
Representation
Theory
Quantum
Mechanics
But first, we need a few lemmas…
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Lemmas
Lemma 1. Symmetry operations generate eigenfunctions.
Suppose H 𝜑 = E𝜑.
If 𝑅𝑎 is a symmetry operation of H, then
H (𝑅𝑎 𝜑) = E (𝑅𝑎 𝜑)
Lemma 2. Symmetry operations phase shift 𝝋 for non-degenerate E.
Suppose E is non-degenerate, then
𝑅𝑎 𝜑 = 𝑒 𝑖𝜃𝛼 𝜑
KRISTINA CHANG
for some complex number 𝑒 𝑖𝜃𝛼 .
REPRESENTATION THEORY IN QUANTUM MECHANICS
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The “Bridging Theorem”:
non-degenerate case
Hamiltonian eigenfunctions form a basis for irreducible representations.
 Suppose {𝑹𝜶 } is the group of the Hamiltonian H.
 Suppose H𝝋= E𝝋, where E is non-degenerate. (We will consider the
degenerate case separately)
 Consider the set of complex numbers
{ 𝑒 𝑖𝜃𝛼 | 𝑅𝛼 𝜑 = 𝜑𝑒 𝑖𝜃𝛼 where 𝑅𝛼 is a symmetry op.}
Th 6.3a.
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{ 𝒆𝒊𝜽𝜶 }
{ 𝒆𝒊𝜽𝜶 } is an irreducible representation of 𝑹𝜶 .
REPRESENTATION THEORY IN QUANTUM MECHANICS
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Hamiltonian eigenfunctions form a basis for irreducible representations.
{ 𝒆𝒊𝜽 } is an
irreducible representation of
The “Bridging
Theorem”
Th 6.3a.
𝜶
𝑹𝜶 .
Proof. 𝑓: {𝑅𝛼 } → GL(1,C),
𝑓 𝑅𝛼 = 𝑒 𝑖𝜃𝛼 where 𝑅𝛼 𝜑 = 𝜑𝑒 𝑖𝜃𝛼 .
Show 𝑓 is a homomorphism.
By def…
𝑓(𝑅𝑎 𝑅𝑏 ) 𝜑 = 𝑒 𝑖𝜃𝑎𝑏 𝜑 = 𝑅𝑎 𝑅𝑏 𝜑 = 𝑅𝑎 𝑒 𝑖𝜃𝑏 𝜑 = 𝑒 𝑖𝜃𝑎 𝑒 𝑖𝜃𝑏 𝜑 = 𝑓(𝑅𝑎 )𝑓(𝑅𝑏 )𝜑
Since, 𝑓(𝑅𝑎 𝑅𝑏 ) = 𝑓(𝑅𝑎 )𝑓(𝑅𝑏 ), the image set { 𝒆𝒊𝜽𝜶 } is a representation.
Since it is 1-dimensional, it is irreducible.
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REPRESENTATION THEORY IN QUANTUM MECHANICS
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The “Bridging Theorem”:
degenerate case
Suppose H𝜑 = E𝜑, and that E is n-degenerate.
Let {R∝} be the group of the Hamiltonian H.
Consider the set of eigenfunctions { Ra 𝜑 | Ra∈ {R∝} } of E.
Def. If { Ra 𝜑 | Ra ∈ {R∝} } contains all n distinct eigenfunctions of E, then its
eigenfunctions are said to be normal degenerate.
Repeated application of Ra on 𝜑  complete set of normal degenerate 𝜑’s
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REPRESENTATION THEORY IN QUANTUM MECHANICS
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The “Bridging Theorem”:
degenerate case
 Suppose {𝑹𝜶 } is the group of the Hamiltonian H.
 Suppose E is an eigenvalue of H with n normal degenerate eigenfunctions
𝝋 = (φ1 , φ2 , … , φ𝑛 ).
 For a given symmetry operation 𝑅𝑎 …
Let Γ 𝑅𝑎 be the nxn matrix which satisfies
𝑅𝑎 𝝋 = 𝝋 Γ 𝑅𝑎
Th 6.3b. The set of nxn matrices { 𝚪 𝑹𝜶 } is an n-dimensional
irreducible representation of {𝑹𝜶 }.
KRISTINA CHANG
REPRESENTATION THEORY IN QUANTUM MECHANICS
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The “Bridging Theorem”
Th 6.3b. The set of nxn matrices { 𝚪 𝑹𝜶 } is an n-dimensional irreducible
representation of {𝑹𝜶 }.
Proof.
(i) Show that { 𝜞 𝑹𝜶 } is a representation.
Show Γ R 𝑎 R 𝑏 = Γ R 𝑎 Γ R 𝑏 .
Recall def. of 𝚪:
𝑹𝒂 𝝋 = 𝝋 𝚪 𝑹𝒂
𝝋 Γ R𝑎 R𝑏 = R 𝑎 R𝑏 𝝋 = R 𝑎 𝝋 Γ R𝑏 = 𝝋 Γ R𝑎 Γ R𝑏 .
Since the matrices in { Γ R 𝛼 } are all nxn, { Γ R 𝛼 } is an n-dimensional representation.
(ii) Show that { 𝜞 𝑹𝜶 } is irreducible.
(given)
KRISTINA CHANG
REPRESENTATION THEORY IN QUANTUM MECHANICS
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In Summary…
Application of
Representation
Theory
Normal degenerate families of eigenfunctions 𝝋
“interact” with the group of the Hamiltonian {𝑹𝜶 }
to generate irreducible rep’s { 𝚪 𝑹𝜶 } .
We will use this result to derive Bloch’s Th:
 form of eigenfunctions for electrons in a crystal
1. find group of the Hamiltonian {𝑹𝜶 }
2. already know irreducible rep’s { 𝚪 𝑹𝜶 }
3. decide what eigenfunctions 𝝋 must look like
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1-D crystal
1-D lattice of positive ions
V(x)
+
a
+
+
+
x
N ions in the unit cell spaced a distance a apart
“Build crystal” by applying periodic boundary condition  N discrete symmetries
Now, H is symmetric under any operator 𝑅𝑎 such that
𝑅𝑎 𝜑 𝑥 = 𝜑 𝑥 + 𝑎 , and (𝑅𝑎 )𝑁 = 𝑅0 = 𝐸
< 𝑹𝒂 > = group of the Hamiltonian
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REPRESENTATION THEORY IN QUANTUM MECHANICS
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Bloch’s Theorem: 1-D
What are the irreducible representations of < 𝑅𝑎 > ?
Recall: For a cyclic group < 𝑹𝒂 > of order N…
where 𝜔 = 𝑒 2𝜋𝑖/𝑁
Table of Irreducible rep’s of < 𝑹𝒂 >
𝐸
𝑅𝑎
Γ1
Γ2
1
1
1
𝑅𝑎 2
1
𝜔
𝜔2
…
…
⋮
Γ𝑁
⋮
1
𝜔𝑁−1
𝜔𝑁−2
…
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…
REPRESENTATION THEORY IN QUANTUM MECHANICS
𝑅𝑎 𝑁−1
1
𝜔𝑁−1
𝜔
(𝑁−1)2
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Bloch’s Theorem: 1-D
Recall: (𝑅𝑎 )𝑛 𝜑 𝑥 = 𝜑 𝑥 + 𝑛𝑎
What does 𝝋𝒏 𝒙 look like?
Suppose 𝜑𝑛 is the eigenfunction which generates { Γ𝑛 }.
Then 𝑅𝑎 𝜑𝑛 𝑥 = 𝜑𝑛 𝑥 Γ𝑛 𝑅𝑎 = 𝜑𝑛 𝑥 𝜔𝑛−1 . We know 𝑅𝑎 𝜑𝑛 𝑥 = 𝜑𝑛 𝑥 + 𝑎 ,
So,
𝜑𝑛 𝑥 𝜔𝑛−1 = 𝜑𝑛 𝑥 + 𝑎 .
Then,
|𝜑𝑛 𝑥 + 𝑎 |2 = |𝜑𝑛 𝑥 𝜔𝑛−1 |2 = |𝜑𝑛 𝑥 |2
Eq. 6.5
General solution to Eq. 6.5:
𝜑𝑛 𝑥 = 𝑒 𝑖𝜃𝑛 (𝑥) 𝑢𝑛 𝑥
Phase function
𝑒 𝑖𝜃𝑛 (𝑥)
Periodic function 𝑢𝑛 𝑥 = 𝑢𝑛 𝑥 + 𝑎
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REPRESENTATION THEORY IN QUANTUM MECHANICS
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Bloch’s Theorem: 1-D
𝜑𝑛 𝑥 = 𝑒 𝑖𝜃𝑛 (𝑥) 𝑢𝑛 𝑥
General solution:
Now apply boundary conditions to 𝜑𝑛 𝑥 …
𝑅𝑎 𝑚 𝜑𝑛 𝑥
= 𝜑𝑛 𝑥 + 𝑚𝑎 = 𝜑𝑛 𝑥 𝜔 𝑚(𝑛−1) ,
→ 𝑒 𝑖𝜃𝑛 (𝑥+𝑚𝑎) 𝑢𝑛 𝑥 + 𝑚𝑎 = 𝑒 𝑖𝜃𝑛 (𝑥) 𝑢𝑛 𝑥 𝜔 𝑚(𝑛−1)
→ 𝑒 𝑖𝜃𝑛 (𝑥+𝑚𝑎) 𝑢𝑛 𝑥 = 𝑒 𝑖𝜃𝑛 (𝑥) 𝑢𝑛 𝑥 𝜔 𝑚(𝑛−1)
→ 𝑒 𝑖𝜃𝑛 (𝑥+𝑚𝑎) = 𝑒 𝑖𝜃𝑛 (𝑥) 𝜔 𝑚(𝑛−1)
→
𝑒 𝑖𝜃𝑛 (𝑥+𝑚𝑎)
=
𝑒 𝑖𝜃𝑛 (𝑥)
𝑒
2𝜋𝑖
𝑁
→ 𝜽𝒏 𝒙 + 𝒎𝒂 = 𝜽𝒏 𝒙 +
KRISTINA CHANG
𝑚(𝑛−1)
𝟐𝝅
𝑵
𝒎(𝒏 − 𝟏)
REPRESENTATION THEORY IN QUANTUM MECHANICS
condition on 𝜽𝒏 (𝒙)
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Bloch’s Theorem: 1-D
Given: 𝜃𝑛 𝑥 = 𝐵 + 𝑘𝑛 𝑥 ,
𝑘𝑛 =
2𝜋
𝑁𝑎
𝑛 − 1 , some constant B
Substitute 𝜃𝑛 𝑥 into general solution for 𝜑𝑛 𝑥 :
𝜑𝑛 𝑥 = 𝑒 𝑖𝜃𝑛 (𝑥) 𝑢𝑛 𝑥
= 𝑒 𝑖(𝐵+𝑘𝑛 𝑥) 𝑢𝑛 𝑥
= 𝑒 𝑖𝑘𝑛 𝑥 𝑢𝑛 𝑥
Bloch functions
𝝋𝒏 𝒙 = 𝒆𝒊𝒌𝒏 𝒙 𝒖𝒏 𝒙
The eigenstates of a 1-D crystal lattice (periodic function modulated by a plane wave)
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REPRESENTATION THEORY IN QUANTUM MECHANICS
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Representation Theory:
We showed that symmetry groups can be represented by
matrices.
Cyclic groups have special sets of irreducible representations.
Summary
Quantum Mechanics:
Hamiltonian belong to symmetry groups.
We can map eigenfunctions (wavefunctions) to irreducible
representations in the Hamiltonian.
This helps us to solve for eigenfunctions (wavefunctions) of a
1-D crystal with only a knowledge of Hamiltonian symmetry.
31
Acknowledgements
Professor Naimi
Friends
Faculty
Professor Buckmire
All of you!
References
[1] Vedensky, Dimitri. “Group Theory: Course Notes” (2001).
http://www.cmth.ph.ic.ac.uk/people/d.vvedensky/courses.html
[2] Cotton, Albert. Chemical Applications of Group Theory (2003). John Wiley and Sons, Inc.
[3] Saracino, Dan. Abstract Algebra: A First Introduction (2008). Waveland Pr, Inc.
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Bloch’s Theorem: 1-D
What does 𝜃𝑛 (𝑥) look like? We know…
𝜽𝒏 𝒙 + 𝒎𝒂 = 𝜽𝒏 𝒙 +
Cute trick:
𝟐𝝅
𝑵
𝒎 𝒏−𝟏
Eq. 6.10
If we hold x and n constant, 𝜽𝒏 looks like a linear function of m.
So 𝜽𝒏 is a linear function of am + 𝑥.
So we can write
𝜽𝒏 𝒙 = 𝑩 + 𝑨𝒙
𝜃𝑛 𝑥 = 𝐴 am + 𝑥 + 𝐵
 Substitute into Eq. 6.10:
𝐵 + 𝐴 𝑥 + 𝑚𝑎 = 𝐵 + 𝐴𝑥 +
Solve for A:
KRISTINA CHANG
𝐴=
2𝜋
𝑁𝑎
𝑛−1
2𝜋
𝑚 𝑛−1
𝑁
= 𝑘𝑛
REPRESENTATION THEORY IN QUANTUM MECHANICS
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