Subatomic Physics

7.1. Conserved Quantities and Symmetries 199
How Can Conserved Quantities Be Found? After resolving the question as
to when an observable is conserved, we attack the more physical problem: How can
conserved quantities be found? The direct approach, writing down H and inserting
all observables into the commutator, is usually not feasible because H is not fully
known. Fortunately, H does not have to be known explicitly; a conserved observable
can be found if the invariance of H under a symmetry operation is established. To
define symmetry operation, we introduce a transformation operator U. U changes
a wave function ψ(x,t) into another wave function ψ ′ (x,t):
ψ ′ (x,t)=Uψ(x,t). (7.7)
Such a transformation is admissible only if the normalization of the wave function
is not changed:
�
�
�
d 3 xψ ∗ ψ =
d 3 x(Uψ) ∗ Uψ =
d 3 xψ ∗ U † Uψ.
The transformation operator U consequently must be unitary, (3)
U † U = UU † = I. (7.8)
U is a symmetry operator if Uψ satisfies the same Schrödinger equation as ψ. From
i� d(Uψ)
dt
= HUψ it follows that i�dψ
dt = U −1 HUψ,
where U is assumed to be time independent and where U −1 is the inverse operator.
Comparison with Eq. (7.1) gives
H = U −1 HU = U † HU or HU − UH ≡ [H, U] =0. (7.9)
The symmetry operator U commutes with the Hamiltonian.
Comparison of Eqs. (7.5) and (7.9) shows the way to find conserved observables.
If U is Hermitian, it will be an observable. If U is not Hermitian, a Hermitian
operator can be found that is related to U and satisfies Eq. (7.5). Before giving
an example of such a related operator, we recapitulate the essential facts about the
operators F and U.
3 Notation and definitions: If A is an operator, the Hermitian adjoint operator A † is defined
by �
d 3 x(Aψ) ∗ �
φ = d 3 xψ ∗ A † φ.
The operator A is Hermitian if A † = A; it is unitary if A † = A−1 or A † A = 1. Unitary operators
are generalizations of eiα , the complex numbers of absolute value 1 (Merzbacher, Chapter 14).
Notation: If A is a matrix with elements aik,A∗ with elements a∗ ik is the complex conjugate
matrix. Ã with elements aki is the transposed matrix. A † with elements a∗ ki is the Hermitian
conjugate (H.C.) matrix. (AB) † = B † A † . I is the unit matrix. The matrix F is called Hermitian
if F † = F .ThematrixUis unitary if U † U = UU † = I.