Lecture 18 8.

321 Quantum Theory I, Fall 2017 79

Lecture 18 (Nov. 13, 2017)

18.1 Symmetries in Quantum Mechanics
A symmetry is a physical operation we can perform on the system that leaves the physics unchanged.
As an example, consider a free particle,
p2
Hfree = . (18.1)
2m
This Hamiltonian does not depend on position, so we can translate x → x + a. Under this transfor-
mation, Hfree is unchanged and [x + a, p] = i~, implying that both H and the commutation algebra
are unchanged. Thus, translation is a symmetry of the free-particle system. This Hamiltonian has
several other symmetries. Inversion (or parity), given by x → −x, p → −p, preserves [x, p] = i~ and
Hfree , and so is a symmetry. Time reversal, which sends t → −t, is also a symmetry; we will discuss
its realization in quantum mechanics later. The system is also invariant under Galilean transfor-
mations, even though Galilean boosts p → p + mv0 seem to change the Hamiltonian. Galilean
boosts change the phase in the path integral by something that does not depend on the path, so
the probabilities do not change, but the amplitudes do; this example is more subtle, and so we will
stick to discussions of the other symmetries of the system.

18.1.1 Symmetry Transformations

We define a symmetry operation as a linear transformation U : |ai → |a0 i such that all results of
measurement are preserved. This means that
2
hb0 |a0 i = |hb|ai|2 (18.2)

for all a, b ∈ H. From the definition of the primed kets, this gives
2
hb|U † U |ai = |hb|ai|2 . (18.3)

Wigner’s theorem tells us that if U satisfies this condition, then U must be unitary or anti-unitary;
an anti-unitary operator U˜ is one that satisfies

Ũ |ai = KU |U i = (U |ai)† , (18.4)

where K is the complex conjugation operator, and U is some unitary operator. If we assume that
U is linear or anti-linear, meaning that
!
X X
U ca |ai = c∗a U |ai , (18.5)
a a

then we can show Wigner’s theorem fairly simply (this will appear on your homework). Wigner’s
theorem can be proved without making this assumption, but the proof is more subtle.

18.1.2 Continuous Symmetries and Conservation Laws

Consider a symmetry operation described by a unitary operator U . This is a symmetry of the
Hamiltonian if H is unchanged by the action of U , i.e.

H = U † HU , (18.6)
Lecture 18 8.321 Quantum Theory I, Fall 2017 80

i.e., [H, U ] = 0. If the symmetry transformation can be continuously built up as a series of

infinitesimal transformations starting from the identity operator 1, then we call it a continuous
symmetry. Translation and rotation are examples of continuous symmetries: any translation or
rotation can be built up from the identity by doing a sequence of infinitesimal translations or
rotations. Symmetries that cannot be built up in this way are called discrete symmetries. Parity
is an example of a discrete symmetry (in even dimensions, we can construct parity as a rotation,
but it is nonetheless possible that the Hamiltonian is symmetric under parity but not rotations, so
parity is still discrete).
Let us now consider continuous symmetries. All of the information about a continuous symmetry
is contained in its infinitesimal form, because we can always build up a continuous transformation
from its infinitesimal version. As we have seen previously, an infinitesimal unitary operator can be
expressed in terms of a Hermitian operator: if we expand an infinitesimal unitary operator U as
i
G + O 2 ,


U =1− (18.7)
~
then the statement U † U = 1 implies G† = G, i.e., G is Hermitian. The statement that U is a
symmetry is U † HU = H, i.e.,
   
i 2

i 2


1+ G+O  H 1− G+O  =H, (18.8)
~ ~

which gives us [G, H] = 0. Because G is Hermitian, it corresponds to some observable, and we now
have
dG 1
= [G, H] = 0 , (18.9)
dt i~
so we see that G is conserved. This leads us to the following general statement:

Theorem 4 (Noether’s Theorem). For every continuous symmetry of the Hamiltonian in quan-
tum mechanics, there is a corresponding conserved quantity. Conversely, if some observable G is
conserved, then [G, H] = 0, and we can define unitary operators

U (θ) = eiθG/~ , (18.10)

which will satisfy U (θ)† HU (θ) = H, showing that U (θ) is a continuous symmetry of the Hamilto-
nian.

18.1.3 Translations
As an example, consider again translation, x → x + a. The translation operator is

T (a) = e−ipa/~ . (18.11)

This satisfies (T (a))† = (T (a))−1 , i.e., the translation operator is unitary. It has the following
properties:

• (T (a))−1 = T (−a),

• T (a0 )T (a00 ) = T (a0 + a00 ),

• T † (a)xT (a) = x + a.
Lecture 18 8.321 Quantum Theory I, Fall 2017 81

For infinitesimal a, we can expand the translation operator as

ia
T (a) ≈ 1 − p , (18.12)
~
where p is the Hermitian generator of T (a). If a system has translation symmetry, then momentum
is conserved, from Noether’s theorem. In d dimensions, we generalize to translations xi → xi + ai ,
with i = 1, . . . , d. Then, the unitary translation operator is
Y Y
T ({ai }) = Ti (ai ) = e−ipi ai /~ . (18.13)
i i
It is a geometric fact that translations in different directions commute, [Ti (ai ), Tj (aj )] = 0, for all
i, j. This tells us that the Hermitian generators commute, [pi , pj ] = 0, for all i, j. We can define
momentum as the observable that corresponds to the Hermitian generator of translations; then
taking a to be infinitesimal in the identity
eipa/~ xe−ipa/~ = x + a (18.14)
gives us the commutation relation [x, p] = i~.

18.1.4 Time Translations

For a closed system, time translation is a symmetry, and the corresponding unitary operator is the
time-evolution operator,
U (t) = e−iHt/~ . (18.15)
The Hamiltonian H is the generator of this symmetry, and so the fact that the system is invariant
under time translation implies the conservation of energy,
dH
= 0. (18.16)
dt

18.1.5 Rotations
If we have a d-dimensional system with position coordinate xi , then a rotation is given by
d
X
xi → x0i = Rij xj = Rij xj , (18.17)
j=1

such that the scalar product is preserved,

x · y = x0 · y 0 . (18.18)
Here we are using the Einstein summation convention that repeated indices are summed over. The
preservation of the scalar product can be written as
Rij Rik xj yk = xi yi , (18.19)
which implies
Rij Rik = δjk , (18.20)
RRT
i.e., = 1. That is, R is an orthogonal matrix.
The orthogonality condition, RRT = 1, implies that (det R)2 = 1, meaning that det R = ±1.
We see then that there are two classes of rotations. Matrices with det R = −1 are not continuously
connected to the identity; they involve inversions xi → −xi combined with an ordinary (orientation-
preserving) rotation. (Note that in even dimensions, parity is orientation-preserving.) For now,
we will stick to considering rotations R that have det R = +1. Each rotation R is represented in
quantum mechanics by a unitary operator D(R) that acts on the system’s Hilbert space.
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8.321 Quantum Theory I

Fall 2017

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