Download Chapter 3 Symmetry in quantum mechanics

Chapter 3
Symmetry in quantum mechanics
In this chapter we want to establish the importance of symmetry considerations in quantum mechanics and discuss the connection between
- symmetries;
- degeneracies;
- conservation laws.
3.1
3.1.1
Symmetries, conservation laws, degeneracies
Symmetries in classical physics
In classical mechanics, one usually considers the Lagrange formulation defined in terms of
the Lagrangian L, which is a function of the generalized coordinates, qi , and the generalized
velocities, q˙i , of the system.
If, for instance, the Lagrangian L remains unchanged under displacements,
qi → qi + δqi ,
then
∂L
= 0,
∂qi
which implies that:
dpi
=0
dt
since
d
dt
∂L
∂ q˙i
−
∂L
=0
∂qi
and
pi =
(3.1)
∂L
∂ q˙i
Equation (3.1) means that pi is a conserved quantity.
In the Hamiltonian formulation,
H(pi , qi ),
65
(canonical momentum).
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CHAPTER 3. SYMMETRY IN QUANTUM MECHANICS
the Hamilton equations are given by:
ṗi = −
and
∂H
,
∂qi
q˙i =
∂H
,
∂pi
d
pi = 0
dt
whenever
∂H
= 0.
∂qi
In other words, if the Hamiltonian doesn’t explicitly depend on qi (which is equivalent to
saying that H is unchanged under qi → qi + δqi ), we have a conserved quantity, in this
case, the momentum pi .
3.1.2
Symmetries in quantum mechanics
In quantum mechanics:
(i) we associate a unitary operator Ô to a transformation that conserves probability.
For instance, a rotation is described by a unitary operator. This operator is often
called a symmetry operator ;
(ii) we classify symmetries as continuous (rotation, translation) and discrete (parity,
lattice translations, time reversal).
(ii) symmetry operations that differ infinitesimally from the identity transformation
(continuous symmetries) are written as:
Ô = 1 −
iε
Ĝ,
~
(3.2)
where Ĝ is the hermitian generator of the symmetry operator we are describing.
(Recall what we did in Chapter 2, where the angular momentum is the generator of
the rotation operator. The same holds for the translation operator).
Translations in quantum mechanics. Translation operation
Let us assume that we start with a state which is well localized around ~r0 . We define an
operation that changes this state into another well-localized state around ~r0 + d~r0 with
everything else unchanged. Such an operation is an infinitesimal translation by d~r0 . We
denote the corresponding operator by T̂ (d~r0 ):
T̂ (d~r0 ) |~r0 i = |~r0 + d~r0 i
From expression (3.3), it is clear that |~r0 i is not an eigenstate of T̂ .
Properties of T̂ (d~r0 ):
(3.3)
3.1. SYMMETRIES, CONSERVATION LAWS, DEGENERACIES
67
1. unitarity (since it has to fulfill probability conservation):
hα|αi = hα| T̂ † (d~r0 )T̂ (d~r0 ) |αi
⇒
T̂ † (d~r0 )T̂ (d~r0 ) = 1;
2. T̂ (d~r1 )T̂ (d~r0 ) = T̂ (d~r1 + d~r0 );
3. a translation in the opposite direction must be the same as the inverse of the original
translation:
T̂ (−d~r0 ) = T̂ −1 (d~r0 );
4.
lim T̂ (d~r0 ) = 1,
d~
r0 →0
identity operator;
5. a translation of a quantum state in position representation is:
D
E
~r|T̂ (dr~0)|Ψ = h~r − dr~0 |Ψi
6. the infinitesimal translation operator can be expressed as:
T̂ (d~r0 ) = 1 − iK̂ · d~r0 ,
(3.4)
where (Kx , Ky , Kz ) are hermitian operators. In order to show that, it is enough to
see that equation (3.4) fulfills the properties 1 to 4 (exercise).
Analogous to rotations and angular momenta as generators of rotations, K̂ is the generator
of an infinitesimal translation. K̂ is related to the momentum operator, ~p, as
K̂ =
p~
.
~
(To see the proof: Sakurai “Modern Quantum Mechanics“, page 45).
Ĝ as a constant of motion
Now, let us go back to our general symmetry operator Ô (Eq. 3.2). We assume that the
Hamilton operator, Ĥ, describing the quantum system is invariant under Ô. In this case,
Ô † Ĥ Ô = Ĥ,
which is equivalent to
h
i
Ĝ, Ĥ = 0.
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CHAPTER 3. SYMMETRY IN QUANTUM MECHANICS
In the Heisenberg representation, the corresponding equation of motion for Ĝ is (provided
that Ĝ doesn’t explicitly depend on time):
i
dĜ
1 h
Ĝ, Ĥ
=
dt
i~
→
dĜ
= 0,
dt
what implies that Ĝ is a constant of motion. Then, for instance:
- if Ĥ is invariant under translation, then the momentum is a constant of motion;
- if Ĥ is invariant under rotation, then the angular momentum is a constant of motion.
In order to see that, let us consider that at t0 the system is in an eigenstate of Ĝ, |g, t0i,
with an eigenvalue g. The state at a later time, t, will be obtained by applying the
time-evolution operator, U(t, t0 ):
|g, t0; ti = U(t, t0 ) |g, t0i .
The new state, |g, t0 ; ti, should be an eigenstate of Ĝ as well, with the same eigenvalue g
since
i
h
Ĝ, Ĥ = 0
h
i
→
Ĝ, U(t, t0 ) = 0
→
Ĝ [U(t, t0 ) |gi] = U(t, t0 )G |gi = g [U(t, t0 ) |gi] .
Degeneracies
The question of degeneracy in quantum mechanics has important implications on the
properties of the quantum system.
Let us consider that the Hamiltonian of a quantum system commutes with the symmetry
operator Ô:
h
i
Ĥ, Ô = 0.
If |ni is an eigenstate of Ĥ,
Ĥ |ni = En |ni ,
then Ô |ni is an eigenstate of Ĥ as well since:
Ĥ Ô |ni = Ô Ĥ |ni = En Ô |ni .
Let us assume now that |ni and Ô |ni are different eigenstates with the same energy En .
In this case, the two states are degenerate.
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3.2. DISCRETE SYMMETRIES: PARITY
We consider the special case of rotations: Ô = R̂. If the Hamiltonian is rotationally
invariant,
i
i
h
i
h
h
ˆ Ĥ = 0, Jˆ2 , Ĥ = 0,
J,
R̂, Ĥ = 0 →
we can define a set of eigenstates of {H, J 2 , Jz } characterized by the quantum numbers
n, j, m.
From what we have seen above, all states of the form:
R̂ |n; j, mi
have the same energy. Since usually under rotation all m-values get mixed, R̂ |n; j, mi is
a linear combination of the 2j + 1 independent states:
X
(j)
R̂ |n; j, mi =
|n; j, m′ i Cm′ m .
m′
The degeneracy of R̂ |n; j, mi is (2j + 1)-fold, which is equal to the number of possible
m-values. This can also be seen from the fact that [J± , H] = 0 and therefore |n; j, mi for
m = −j, . . . , j have all the same energy.
Application of this degeneracy. As an illustration:
Example 1: we consider an electron under the influence of a potential
~ · S,
~
V (r) + ε(r)L
~ ·S
~ being the spin-orbit coupling.
with L
~ ·S
~ (scalar product) are rotationally invariant. Therefore, we will have for
Both r and L
~ + S.
~ If we apply now an
each atomic level, n, a (2j + 1)-fold degeneracy, with J~ = L
electric field in the z-direction, the rotational symmetry becomes broken, and states with
different m-values acquire different energies.
Example 2: two spin 1/2 particles described by the Heisenberg interaction in the
presence of an external magnetic field,
X
~1 · S
~2 + B
Siz .
H = JS
i=1,2
~1 · S
~2 .
→ Show the breaking of the triplet degeneracy for the interaction S
3.2
Discrete symmetries: parity
Rotations and translations are continuous symmetry operators, i.e., they describe operations that can be obtained by applying successively infinitesimal operations. There are
though symmetry operations that cannot be obtained in this way, for instance:
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CHAPTER 3. SYMMETRY IN QUANTUM MECHANICS
- parity or space inversion
- lattice translation
- time reversal
They all are discrete symmetries.
Parity operation changes a right-handed (RH) system to a left handed (LH) system:
We define the parity operator, Π (it can also be denoted as P ), in such a way that:
Π†~r Π = −~r
or
~r Π = −Π~r.
Properties of Π:
1. it is unitary,
ΠΠ† = 1,
and therefore Π and ~r anticommute:
{Π, ~r} = 0.
Note:
n
o
Â, B̂ denotes anticommutator: ÂB̂ + B̂ Â.
2. an eigenfunction of the position operator, |~r0 i, transforms under Π as:
Π |~r0 i = eiδ |−~r0 i ,
where eiδ is a phase factor (δ is real).
Proof : since
~r Π |~r0 i = −Π~r |~r0 i = (−~r0 )Π |~r0 i ,
(3.5)
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3.2. DISCRETE SYMMETRIES: PARITY
Π |~r0 i is an eigenstate of ~r with eigenvalue −~r0 . This means that, up to a phase
factor, the state Π |~r0 i should be the same as |−~r0 i. By convention, we take the
phase factor eiδ = 1 . With this choice of eiδ , eq. (3.5) becomes:
Π |~r0 i = |−~r0 i ,
Π2 |~r0 i = |~r0 i
→
Π2 = 1 , identity operator.
3. since Π2 = 1, Π is also hermitian,
Π−1 = Π† = Π,
with eigenvalues +1 and −1.
4. the momentum operator, ~p, is also, as ~r, odd under parity:
Π† ~p Π = −~p .
Proof : the state |~pi in the position representation is:
Z
1
d3 r ei~p·~r/~ |~ri .
|~pi =
(2π~)3/2
We act on |~pi by Π:
Z
1
Π |~pi =
d3 r ei~p·~r/~ Π |~ri
(2π~)3/2
Z
1
d3 r ei~p·~r/~ |−~ri
=
(2π~)3/2
Z
1
′
=
d3 r ′ e−i~p·~r /~ |~r ′ i = |−~pi .
3/2
(2π~)
and therefore,
Π |~pi = |−~pi .
Then, for the operator p~ we have (in analogy to ~r):
Π† ~p Π = −~p;
{Π, ~p} = 0.
r
.
This can also be seen from ~p = m d~
dt
~ under parity:
5. the orbital angular momentum, L
h
i
~ = 0.
Π, L
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CHAPTER 3. SYMMETRY IN QUANTUM MECHANICS
~ L
~ = ~r × p~, with ~r and p~ being
This can be shown by considering the definition of L,
~
odd in parity, L is even in parity:
~ = LΠ.
~
ΠL
~ is the generator of spatial rotations. The 3 × 3
or by considering the fact that L
(parity)
orthogonal matrices, R
and R(rotation) , commute:
R(parity) R(rotation) = R(rotation) R(parity) ,
where R(parity) (or parity inversion) is

R(parity)

−1 0
0
=  0 −1 0  .
0
0 −1
This commutation relation should be fulfilled by the corresponding operators, Π
and R̂:
ΠR̂ = R̂Π,
with
i ~
R̂ = 1 − dαL
· ~u.
~
Then,
i ~
i ~
Π 1 − dαL · ~u = 1 − dαL · ~u Π
~
~
→
~ = LΠ
~
ΠL
→
~ =0
[Π, L]
or
~ = L.
~
Π† LΠ
~ Both spin and L
~ are even
This can be generalized to any angular momentum J.
operators under parity:
~ =S
~.
Π† SΠ
Definitions:
-polar vectors: vector operators that are odd under parity, like
~r, ~p;
-axial vectors or pseudovectors: vector operators that are even under parity,
like
~
J;
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3.2. DISCRETE SYMMETRIES: PARITY
-scalar operators: under parity they transform as ordinary scalars, like
~ ·S
~
L
→
~ · SΠ
~ =L
~ · S;
~
Π−1 L
-pseudoscalar operators: scalar operators which under parity transform as:
Π−1 ÂΠ = −Â.
e.g.,
→
3.2.1
~ · ~r
S
~ · ~r Π = −S
~ · ~r.
Π−1 S
Wavefunctions under parity
→ h~r |Ψi = Ψ(~r)
→ h~r| Π |Ψi = h−~r |Ψi = Ψ(−~r)
If |αi is an eigenstate of Π, then
In the position representation:
Π |αi = ± |αi .
h~r| Π |αi = ± h~r |αi
and h~r| Π |αi = h−~r |αi
h~r |αi = ± h−~r |αi .
Therefore, the eigenfunctions of parity can be classified as even and odd :
α(−~r) = α(~r) even
α(−~r) = −α(~r) odd
Example of wavefunctions with definite parity:
h
i
~ Π = 0, the eigenfunctions of L
~ must have definite parity.
(1) Since L,
In spherical coordinates the transformation ~r → −~r is fulfilled by (see Fig. 3.1):
r → r
θ → π−θ
φ → π+φ
Using relations (3.6) and the definition of Ylm (θ, φ),
s
(2l + 1)(l − m)! m
Ylm (θ, φ) = (−1)m
Pl (cos θ)eimφ ,
4π(l + m)!
it can be shown that:
Ylm (r̂) = (−1)l Ylm (−r̂) .
(3.6)
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CHAPTER 3. SYMMETRY IN QUANTUM MECHANICS
Figure 3.1: Transformation ~r → −~r
(2) Parity properties of energy eigenstates: if [H, Π] = 0, and Ψn is a non-degenerate
eigenstate of H with eigenvalue En ,
H |Ψn i = En |Ψn i ,
then |Ψn i is also an eigenstate of the parity operator.
For instance, in the case of the harmonic oscillator,
1
, n̂ = a† a,
Ĥ = ~ω n̂ +
2
we have that
and
h
i
Ĥ, Π = 0 (shown in QM I)
Ψn (x) = (−1)n Ψn (−x) .
Remark : for the previous statement to be valid, it is important that the eigenstates
of H are non-degenerate , otherwise is does not hold. For instance, consider the
hydrogen atom in the non-relativistic limit,
H = T + V (r),
[H, Π] = 0,
the eigenstates of H are defined by |n; lmi. Considering the two states
2s : |2; 00i
and
2p−1 : |2; 1 − 1i ,
one sees that an energy state cp |2p−1 i + cs |2si is not a parity eigenstate.
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3.2. DISCRETE SYMMETRIES: PARITY
3.2.2
Example: double-well potential
Let us consider a symmetrical double-well potential (see Fig. 3.2):
Figure 3.2: Double-well potential
The solutions for a system in such a potential are given by sin and cos in the classically
allowed region and sinh and cosh in the classically forbidden region and they are matched
at the position where the potential is discontinuous.
The two lowest-lying states can be described by a symmetrical state |Si and an antisymmetrical state |Ai (see Fig. 3.3):
(a) Symmetrical state |Si
(b) Antisymmetrical state |Ai
Figure 3.3: Double-well solutions
Since [H, Π] = 0, these states are simultaneous eigenstates of the hamiltonian and the
parity operator and the symmetrical state has lower energy:
EA > ES .
If the middle barrier gets thinner, the difference EA − ES gets smaller. We define the
right-handed state as:
1
|Ri = √ (|Si + |Ai) ,
(3.7)
2
and the left-handed state as:
1
|Li = √ (|Si − |Ai)
2
(3.8)
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CHAPTER 3. SYMMETRY IN QUANTUM MECHANICS
The names that we have given to the above defined states reflect the fact that the state
(3.7) concentrates mostly on the right-hand side well, and the state (3.8) concentrates
mostly on the left-hand side well.
Note that |Ri and |Li are NOT parity eigenstates. They are non-stationary states:
1
|R, t0 = 0; ti = √ e−iES t/~ |Si + e−iEA t/~ |Ai
2
1 −iES t/~
= √ e
|Si + e−i(EA −ES )t/~ |Ai .
2
the system is in the left-handed state |Li and for t = T the
For t = T /2 ≡ 2(EA2π~
−ES )
system is in the right-handed state |Ri. Thus, as a function of time, the system shows an
oscillatory behavior between |Ri and |Li with angular frequency:
ω=
(EA − ES )
2π
=
.
T
~
A particle initially confined in the right-hand side well can tunnel through the classically
forbidden region to the left-hand side well and back with angular frequency ω.
Let us now consider an infinitely high barrier (see Fig. 3.4)
Figure 3.4: Infinite barrier potential
In this case, there is no tunneling, and the |Si and |Ai states are degenerate (see Fig.
3.5).
They are energy eigenstates, BUT they are not parity eigenstates; the |Li and |Ri states
are also eigenstates of the Hamiltonian, but the system remains forever in |Li or in |Ri.
The ground state is asymmetrical even though the Hamiltonian is symmetrical under
space inversion. This is called broken symmetry related to degeneracy.
One example of broken symmetry in nature is ferromagnetism.
The Hamiltonian for a
P ~~
system of Fe atoms is rotationally invariant (H = ij J Si Sj ), but the ferromagnet has
a definite direction in space. The ground state is degenerate. The infinite number of
degenerate ground states of the ferromagnet are not rotationally invariant since the spins
are aligned along some definite direction (arbitrary).
3.2. DISCRETE SYMMETRIES: PARITY
77
Figure 3.5: Degenerate states
Ammonia molecule NH3
As another example of a system with broken symmetry, let us consider the ammonia
molecule NH3 . In the ammonia molecule, the N atom can be either up or down as shown
in Fig. 3.6:
Figure 3.6: Possible configurations for NH3
The three H atoms are in the corners of an equilateral triangle. The molecule rotates
along the axis as shown in the figure.
The up/down positions of the NH3 molecule are analogous to the R/L states in the
double-well potential. The system oscillates between the up and down positions, the
parity and energy eigenstates, |Si and |Ai, are a superposition of the up/down states,
and the oscillation frequency is
ω=
(EA − ES )
= 24000 MHz,
~
λ = 1 cm
(microwave region).
Due to this two-state configuration, NH3 is used in maser (Microwave Amplification by
Stimulated Emission of Radiation) physics.
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CHAPTER 3. SYMMETRY IN QUANTUM MECHANICS
The excess energy of |Ai is given up to the time dependent potential as |Ai turns to |Si,
and the radiation (microwave) field gains energy (QM I).
Some organic molecules, like sugar or aminoacids, are of R-type or L-type only. In
those cases, the oscillation time between |Ri and |Li is very large: 104 − 106 years.
Therefore, |Ri-type molecules remain right-handed and |Li-type molecules remain lefthanded. When synthesized in the lab, these organic molecules appear with equal mixtures
of |Ri and |Li. But nature gives preference to one configuration. Why this is the case, is
still unknown.
Parity selection rules
Let us consider that the states |αi and |βi are parity eigenstates:
Π |αi = εα |αi ,
Π |βi = εβ |βi ,
where εα , εβ = ±1.
It follows that hβ| ~r |αi = 0 always holds except for the case when:
εα = −εβ ,
i.e., when the states |αi and |βi are of opposite parity.
Proof :
hβ| ~r |αi = hβ| Π−1 Π~
Π−1} Π |αi
| r{z
−~
r
= εα εβ (−1) hβ| ~r |αi
⇒ εα εβ = −1 or hβ| ~r |αi = 0
This is important for radiative transitions. If Ψβ and Ψa are of the same parity,
Z
Ψ∗β ~r Ψα d3 r = 0,
3.3. DISCRETE SYMMETRY: LATTICE TRANSLATION
79
which is a mathematical expression of the Laporte and Wigner rules allowing radiative
transitions to take place only between states of opposite parity. The electric dipole term
~ ·~r. If a Hamiltonian H is invariant under parity,
in a multipole expansion is of the form E
the non-degenerate states cannot possess a permanent dipole moment,
hΨn | ~r |Ψn i = 0.
~ ·~r, . . .) has non-vanishing matrix
Generally, any operator that is odd under parity (~p, ~r, S
elements only between states of opposite parity.
Weak interaction. The Hamiltonian that describes the weak interaction of elementary
particles is not invariant under parity. Lee and Yang in 1956 proposed that parity, which
had been believed to be an inherent symmetry in all important processes in physics,
was not conserved in weak interactions. Namely, in decay processes the final states are
superpositions of opposite parity states.
3.3
Discrete symmetry: lattice translation
We consider now a periodic potential in one dimension, such that
V (x ± a) = V (x).
For instance, we have to deal with this type of potential if we are interested in the
description of the state of an electron under the potential of a chain of regularly spaced
positive ions (see Fig. 3.7).
Figure 3.7: Periodic 1D potential
In the case that the barrier height between two adjacent lattice sites becomes infinite then
the potential is as shown in Fig. 3.8.
The Hamiltonian of the electron is, in general, not invariant under a translation with l
arbitrary:
T̂ † (l)xT̂ (l) = x + l,
T̂ (l) |xi = |x + li .
However, if l = a, the lattice spacing, the situation changes:
T̂ † (a)V (x)T̂ (a) = V (x + a) = V (x),
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CHAPTER 3. SYMMETRY IN QUANTUM MECHANICS
Figure 3.8: Periodic 1D potential with infinitely high barriers between adjacent sites
and since the kinetic term of the Hamiltonian is invariant under any translation then:
T̂ † (a)H T̂ (a) = H
→
[H, T̂ (a)] = 0.
Properties
(i) The ground state of H for the case when the barrier between sites is infinitely high
will be a state in which the electron is localized at the nth site:
|Ψn i ,
with
H |Ψn i = E0 |Ψn i ,
and in which the wavefunction hx |Ψn i = Ψn (x) is finite only in the nth site. BUT,
any state localized at some other site has the same energy E0 , and, therefore, there
are an infinite number of ground states Ψn , where n goes from −∞ to ∞.
(ii) |Ψn i is not an eigenstate of the translation operator T̂ (a) since
T̂ (a) |Ψn i = |Ψn+1 i ,
even though
[H, T̂ (a)] = 0 and H |Ψn i = E0 |Ψn i .
This is so because we have an infinite-fold degeneracy, which means that the symmetry in which nature manifests itself not necessarily has to coincide with the symmetry
of the energy eigenstates.
(iii) There must be though a basis set of states which are simultaneous eigenstates of H
and T̂ (a). Let us consider the linear combination:
|Ψθ i ≡
+∞
X
n=−∞
einθ |Ψn i ,
with
θ ∈ R,
−π ≤ θ ≤ π.
(3.9)
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3.3. DISCRETE SYMMETRY: LATTICE TRANSLATION
We will show that |Ψθ i is a simultaneous eigenstate of H and T̂ (a). Acting with H
on eq. (3.9), we get:
H |Ψθ i =
+∞
X
n=−∞
einθ H |Ψn i = E0
+∞
X
n=−∞
einθ |Ψn i = E0 |Ψθ i ,
and then with T̂ (a):
T̂ (a) |Ψθ i
=
+∞
X
n=−∞
=
+∞
X
n=−∞
=
einθ T̂ (a) |Ψn i
einθ |Ψn+1 i =|n′ =n+1
e−iθ |Ψθ i .
+∞
X
n′ =−∞
′
ei(n −1)θ |Ψn′ i
We see now that |Ψθ i is an eigenstate of H with eigenvalue E0 , and an eigenstate
of T̂ (a) with eigenvalue e−iθ .
3.3.1
Tight-binding approximation
(iv) Let us consider the case of H when the barrier between two adjacent sites is not
infinite (more realistic situation) (see Fig. 3.7). In this case, one can also construct the
localized state |Ψn i with T̂ (a) |Ψn i = |Ψn+1 i. However, due to the quantum tunneling
effect, the electron will have some probability to go into neighboring lattice sites, so that
hx |Ψn i has a tail extending to sites other than the nth site. Therefore, the diagonal matrix
elements of H in the {|Ψn i} basis are all equal (because of translational invariance),
hΨn | H |Ψn i = E0 ,
but H is not completely diagonal in this basis, i.e., there are non-zero off-diagonal matrix
elements. In solid state, we make an approximation that the only important non-diagonal
matrix elements are those connecting nearest neighbors:
′
n = n or
hΨn′ | H |Ψn i 6= 0 only for
n′ = n + 1,
and this approximation is called the tight-binding approximation.
With the definitions:
hΨn±1 | H |Ψn i = −∆
and
hΨn |Ψn′ i = δnn′ ,
we will have:
H |Ψn i = E0 |Ψn i − ∆ |Ψn+1 i − ∆ |Ψn−1 i
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CHAPTER 3. SYMMETRY IN QUANTUM MECHANICS
Though |Ψn i is not an eigenstate of H, one can show that the linear combination |Ψθ i,
|Ψθ i =
+∞
X
n=−∞
einθ |Ψn i ,
which has previously been shown to be an eigenstate of T̂ (a), is an eigenstate of H with
eigenvalue E0 − 2∆ cos θ:
X
X
einθ |Ψn i
H |Ψθ i = H
einθ |Ψn i = E0
X
X
− ∆
einθ |Ψn+1 i − ∆
einθ |Ψn−1 i
X
′
′
= E0 |Ψθ i − ∆
ei(n −1)θ |Ψn′ i − ∆ei(n +1)θ |Ψn′ i
= E0 − ∆ e−iθ + eiθ |Ψθ i
= (E0 − 2∆ cos θ) |Ψθ i .
Since θ ∈ R, we have a continuous distribution of energy eigenvalues between E0 −2∆ and
E0 + 2∆. The discrete energy levels, E0 , form a continuous energy band as ∆ is increased
from zero (Fig. 3.9).
Figure 3.9: Formation of the energy band
3.3.2
Bloch’s theorem
We have that
hx |Ψθ i = Ψθ (x),
and, if T̂ (a) |Ψθ i = e−iθ |Ψθ i,
hx| T̂ (a) |Ψθ i = hx − a |Ψθ i ,
hx − a |Ψθ i = e−iθ hx |Ψθ i .
(3.10)
In order to find θ as a function of a, we set
hx |Ψθ i = eikx uk (x),
with θ = ka,
(3.11)
3.3. DISCRETE SYMMETRY: LATTICE TRANSLATION
83
where uk (x) is a periodic function with period a uk (x − a) = uk (x). Then, substituting
(3.11) in (3.10):
eik(x−a) uk (x − a) = eikx uk (x)e−ika ,
we get the equation verified, which leads us to the following theorem.
Bloch’s theorem: the wavefunction |Ψθ i, which is an eigenstate of T̂ (a), can be written as
a plane wave eikx times a periodic function with periodicity a:
hx |Ψθ i = eikx uk (x).
This theorem holds even when the tight-binding approximation is not anymore valid.
Physical meaning of θ = ka
Bloch’s theorem states that the wave function of the electron is a plane wave characterized
by the propagation wave vector k modulated by a periodic function uk (x). As θ varies
from −π to π, k varies from −π/a to π/a and the energy eigenvalue is a function of k:
E(k) = E0 − 2∆ cos ka .
This is valid within the tight-binding approximation.
The wave vector k fulfills |k| ≤ π/a and describes the Brillouin zone. E(k) is the dispersion relation for the electron states. Since the electron can tunnel, the infinite-fold
degeneracy that we had for the case of infinite height wells is lifted, and the allowed energy values form a continuous band between E0 − 2∆ and E0 + 2∆ (see Fig. 3.10).
Figure 3.10: Energy dispersion in the first Brillouin zone
Considering the case of many electrons moving in a periodic potential and neglecting
Coulomb repulsion among them, we will have that, since the electrons, as fermionic particles, satisfy the Pauli principle, they start filling the band (one electron per site). In the
case of a half-filled band, we will have a metallic state (see Fig. 3.11).
84
CHAPTER 3. SYMMETRY IN QUANTUM MECHANICS
Figure 3.11: Metallic state
3.4
Discrete symmetry: time-reversal
Eugene Wigner (Physics Nobel Prize 1963, together with Maria Goeppert-Mayer and J.
Hans D. Jensen) was the first to discuss this symmetry in 1932.
Let us consider a classical particle that at t = 0 stops and reverses its motion:
p~ |t=0 → −~p |t=0
(a) at t = 0 the particle stops
(b) reverses motion ~p |t=0 → −~
p |t=0
Then, if ~r(t) is a solution of
~ (~r),
m~r¨ = −∇V
~r(−t) is also a solution (as long as the force is non-dissipative), and it is difficult to tell
which sequence, (1) or (2), is the correct one (see Fig. 3.12).
This is not the case if we consider the trajectory of the
particle (in particular, if the particle is an electron) under
~ Microscopically, B
~ is
the influence of a magnetic field B.
produced by moving charges via an electric current; then,
~ B
~ will reverse
if we reverse the current that produces B,
direction.
From the Maxwell equations (in SI unities),
~ ·E
~ = ρ/ǫ0
∇
~ ·B
~ =0
∇
~ ×E
~ = − ∂ B~ ,
~ ×B
~ − ǫ0 µ0 ∂ E~ = µ0~j ∇
∇
∂t
∂t
Figure 3.12: Possible particle trajectory
85
3.4. DISCRETE SYMMETRY: TIME-REVERSAL
it follows that the Lorenz force,
h
i
~ + ~v × B
~ ,
F~ = e E
is invariant under t → −t if
~ →E
~
E
~ → −B
~
B
ρ→ρ
~j → −~j
~v → −~v
In quantum mechanics, if Ψ(~r, t) is a solution of the Schrödinger equation,
∂Ψ
~2 2
i~
= −
∇ + V Ψ,
∂t
2m
Ψ(~r, −t) is not a solution because of the first time derivation, but Ψ∗ (~r, −t) is a solution
since it fulfills the Schrödinger equation:
Ψ(~r, t) = Ψn (~r)e−iEn t/~ ,
Ψ∗ (~r, −t) = Ψ∗n (~r)e−iEn t/~ .
Therefore, time reversal must be related with complex conjugation.
For a spinless system, at t = 0 the wave function is given by
Ψ(~r) = h~r|Ψi ,
and the wave-function for the corresponding time-reversed state is
Ψ∗ (~r) = h~r|Ψi∗ = hΨ|~ri .
3.4.1
Antiunitary transformation
Rotations, translations, and parity are all unitary operators since they keep the scalar
product of two states unchanged:
D
E
hΨ|Φi = hΨ| U † U |Φi = Ψ̃|Φ̃ ,
E
E
where Ψ̃ and Φ̃ are the transformed states through U. For time reversal, one imposes
a weaker requirement:
D
E
Ψ̃|Φ̃ = |hΨ|Φi| ,
which means that either
or
D
E
Ψ̃|Φ̃ = hΨ|Φi∗ = hΦ|Ψi
D
E
Ψ̃|Φ̃ = hΨ|Φi
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CHAPTER 3. SYMMETRY IN QUANTUM MECHANICS
Definition: the transformation
is antiunitary if
and
E
|Ψi → Ψ̃ = Θ |Ψi ,
E
|Φi → Φ̃ = Θ |Φi
D E
Φ̃ Ψ̃ = hΦ |Ψi∗
Θ (c1 |Ψi + c2 |Φi) = c∗1 Θ |Ψi + c∗2 Θ |Φi .
(3.12)
Eq. (3.12) defines Θ as an antilinear operator as well.
An antiunitary operator can be written as
Θ = UK ,
(3.13)
the product of a unitary operator U and a complex-conjugate operator K defined as
1
i
1
i
∗
Kc |Ψi = c K |Ψi ,
f.i., K √ |Sz ; +i ± √ |Sz ; −i → √ |Sz ; +i ∓ √ |Sz ; −i .
2
2
2
2
Note that the effect of K changes with the basis. For instance if the Sz eigenkets (|Sz ; +i,
|Sz ; −i) are used as a base kets in order to represent the Sy states (see above) then K acts
as above and changes the states. However, if we use the Sy eigenkets themselves (|Sy ; ±i,
they do not change under the action of K.
To show that (3.12) fulfills (3.13):
Θ (c1 |Ψi + c2 |Φi) = UK (c1 |Ψi + c2 |Φi)
= c∗1 UK |Ψi + c∗2 UK |Φi
= c∗1 Θ |Ψi + c∗2 Θ |Φi .
3.4.2
Time-reversal operator
We define the time-reversal operator Θ as an antiunitary operator, such that
|Ψi → Θ |Ψi ,
where Θ |Ψi is the time-reversed state of |Ψi, f.i., Θ |~pi = |−~pi (up to a phase factor).
We consider a system described by |Ψi at time t = 0. The system evolves under H, and
at t = δt we have:
iH
|Ψ, t0 = 0; t = δti = 1 −
δt |Ψi .
~
Now, let us first apply Θ at t = 0, and then let the system evolve under H. In this case,
at t = δt we have:
iH
δt Θ |Ψi .
(3.14)
1−
~
87
3.4. DISCRETE SYMMETRY: TIME-REVERSAL
If this evolution of the system is invariant under time reversal, (3.14) should be the same
as
Θ |Ψ, t0 = 0; t = −δti ,
~ In other words,
i.e., when we first consider the state at t = −δt and then reverse p~ and J.
1 − iH
δt Θ |Ψi = Θ 1 − iH
(−δt) |Ψi
~
~
(1)
(2)
Then,
−iHΘ |i = ΘiH |i ,
|i: any ket.
(3.15)
We see from eq. (3.15) that Θ has to be antiunitary. If Θ were unitary, we would have
the equation:
−HΘ = ΘH
and
HΘ |Ψn i = −ΘH |Ψn i = −(En )Θ |Ψn i ,
which would imply that, if |Ψn i is an eigenstate of H with eigenvalue En , then Θ |Ψn i is
an eigenstate of H with eigenvalue −En , what does not make sense.
For instance, if we consider a free particle with energy spectrum
E ∈ [0, +∞),
−E would ∈ (−∞, 0), but there is no state lower than a particle at rest, and, therefore,
the energy spectrum (−∞, 0) does not make sense. From this we conclude that Θ should
be antiunitary,
ΘiH |i = −iΘH |i ,
and eq. (3.15) is then
After canceling the i’s we get:
−iHΘ |i = −iΘH |i .
HΘ = ΘH;
[H, Θ] = 0 ,
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CHAPTER 3. SYMMETRY IN QUANTUM MECHANICS
Note that:
hΨ| Θ |Φi = (hΨ|) (Θ |Φi)
and not
(hΨ |Θ|) · |Φi .
3.4.3
Transformation of operators under time reversal
Consider a linear operator  and states |Ψi and |Φi,
E
Ψ̃ = Θ |Ψi ,
E
Φ̃ = Θ |Φi .
From the antiunitary nature of Θ it follows that
E
D hΦ|  |Ψi = Ψ̃ Θ† Θ−1 Φ̃ .
Proof : we define |γi as
|γi ≡ † |Φi ,
(3.16)
hγ| = hΦ| Â.
Then
D
hΦ| Â |Ψi = hγ |Ψi = Ψ̃ |γ̃i
E
D = Ψ̃ Θ† Φ
E
D = Ψ̃ Θ† Θ−1 Θ Φ
E
D = Ψ̃ Θ† Θ−1 Φ̃ ,
and we get eq. (3.16). If  = † (hermitian), then:
E
D hΦ| Â |Ψi = Ψ̃ ΘÂΘ−1 Φ̃
Definitions:
ΘÂΘ−1 = +Â
ΘÂΘ−1 = −Â
→ observable  is even under time reversal symmetry,
→ observable  is odd under time reversal symmetry.
We substitute these definitions into eq. (3.17):
D E
hΦ| Â |Ψi = ± Ψ̃ Â Φ̃ .
(3.17)
3.4. DISCRETE SYMMETRY: TIME-REVERSAL
For |Ψi = |Φi,
89
D E
hΨ| Â |Ψi = ± Ψ̃ Â Ψ̃ ,
the expectation value taken with respect to the time-reversed state.
Momentum p~: odd
By considering the corrspondence principle it is reasonable to assume that the expectation value of ~p taken with respect to the time-reversed state will be of opposite
sign:
D E
hΨ| p~ |Ψi = − Ψ̃ p~ Ψ̃ ,
therefore we take ~p to be an odd operator with respect to time-reversal symmetry:
Θ~p Θ−1 = −~p .
This has as a consequence,
~p (Θ |~p ′ i) = −Θ~p |~p ′i = (−~p ′ )Θ |~p ′ i ,
i.e., Θ |~p ′ i is a momentum eigenstate of p~ with eigenvalue −~p ′ .
Position ~r: even
Θ~r Θ1 = ~r
Θ |~r ′ i = |~r ′ i
~ odd
Angular momentum J:
To preserve
[Ji , Jj ] = i~ǫijk Jk ,
the angular momentum must be odd under time reversal,
~ −1 = −J,
~
ΘJΘ
~ = ~r × ~p.
which can immediately be seen when J~ = L
3.4.4
Wavefunctions under time reversal
By acting on
|Ψi =
Z
d3 r |~ri h~r |Ψi
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CHAPTER 3. SYMMETRY IN QUANTUM MECHANICS
with Θ:
Θ |Ψi =
=
Z
Z
d3 r Θ |~ri h~r |Ψi∗
d3 r |~ri h~r |Ψi∗
(with the phase convention Θ |~ri = |~ri),
one shows that under time reversal:
Ψ(~r) → Ψ∗ (~r) .
For instance, the spherical harmonics:
Ylm (θ, φ) → Ylm∗ (θ, φ) = (−1)m Yl−m (θ, φ);
therefore,
Θ |l, mi = (−1)m |l, −mi ,
which can be generalized for a state |j, mi:
Θ |j, mi = i2m |j, −mi .
Note that if j if half-integer,
Θ2 |jmi = − |jmi ,
and if j is integer,
Θ2 |jmi = |jmi .
3.4.5
Kramers degeneracy
We consider a charged particle in an external electric field. For a static electric field, the
potential energy is:
V (~r) = eΦ(~r),
where Φ(~r) is the electrostatic potential. Since Φ(~r) is a real function of the time-reversal
even operator ~r, Θ and H commute:
[Θ, H] = 0 .
(3.18)
This relation doesn’t lead though to a conservation law since
ΘU(t, t0 ) 6= U(t, t0 )Θ,
and there is no time-reversal quantum number, but it does lead to such a consequence as
Kramers degeneracy.
3.4. DISCRETE SYMMETRY: TIME-REVERSAL
91
Kramers degeneracy
We assume that [Θ, H] = 0 and H |Ψn i = En |Ψn i. |Ψn i and Θ |Ψn i describe energy
eigenstate and its time-reversed state, respectively, with the same energy eigenvalue since
ΘH |Ψn i = En Θ |Ψn i = HΘ |Ψn i .
We consider now that |Ψn i and Θ |Ψn i represent the same eigenstate. Then, |Ψn i and
Θ |Ψn i can differ only by a phase factor:
Θ |Ψn i = eiδ |Ψn i .
In this case,
→
Θ2 |Ψn i = Θeiδ |Ψn i = e−iδ Θ |Ψn i = e−iδ eiδ |Ψn i
Θ2 |Ψn i = + |Ψn i .
This relation cannot be fulfilled by a system with half-integer j since for those systems
Θ2 = −1, and, therefore, |Ψn i and Θ |Ψn i must correspond to different states with the
same energy: they are degenerate.
Consequences: in a system composed of an odd number of electrons, in an external electric field E each energy level must be at least two-fold degenerate, and this will have
important consequences in the distinct behavior between odd-electron and even-electron
systems.
Kramers was the first to realize this degeneracy when studying the solutions of the
Schrödinger equation, but it was Wigner who showed that this degeneracy is a consequence of time-reversal symmetry.
Magnetic field. If we set a charged particle in a magnetic field, the Hamiltonian will
contain terms of the form:
~ · B,
~
S
~+A
~ · p~
p~ · A
since
H=
~ =∇
~ × A)
~
(where B
i2
1 h
~ + qV (~r)
p~ − eA
2m
(it will be worked out later on).
~ and ~p are odd under time reversal,
Now, since S
ΘH 6= HΘ.
Thus, for instance, for a spin-1/2 system, the spin-up state |+i and its time-reversed
state |−i no longer have the same energy in the presence of an external magnetic field.
Therefore, Kramers degeneracy in a system containing an odd number of electrons is lifted
under application of an external magnetic field.
~ is an external field, it is not affected by the time reversal since time reversal
Note: when B
is only applied to the closed quantum-mechanical system.
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CHAPTER 3. SYMMETRY IN QUANTUM MECHANICS