Notes 21: Symmetries and conservation laws
In classical mechanics, symmetries in the Hamiltonian are associated with conservation laws. For example, if the Hamiltonian is invariant when the coordinate system is rotated about a fixed direction then the component of angular momentum about that direction will be a conserved quantity. The quantum mechanical equivalent is that the expectation value of the component of the angular momentum will be time-independent
Transformations in space Transformations in space include translations, inversions and rotation. The effect of a transformation on a wave function is described by an operator. A one-dimensional translation, in which a function is shifted by a distance a is given by the operator
Taˆ ( )ψψ( x) =′( x) = ψ( x − a). (21.1)
This transformation is illustrated in the figure.
x-a x
The parity operator reflects a function in one dimension about the origin and is defined by
Π=ˆ ψψ( xx) ′( ) =− ψ( x). (21.2)
The effect of the parity operator is shown in the figure below.
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-x x
The translation operator The result of applying the translation operator to a function is given in equation (21.1). An expression for the translation operator can be found by using a Taylor series expansion
n ∞ (−a) d n ˆ ψψ= −= ψ Ta( ) ( x) ( x a) ∑ n ( x). (21.3) n=0 n! dx
The derivative with respect to x is related to the momentum operator by
di = pˆ. (21.4) dx
We see that
n ∞ 1 ia ia Taˆ ( ) =−=−∑ pˆˆexp p . (21.5) n=0 n!
If a is small, then
ia Taˆ ( ) =1. − pˆ (21.6)
A finite translation can be built up from infinitesimal translations. We say that the momentum operator is the generator of translations.
The translation operator is a unitary operator, i.e. its inverse is equal to its adjoint. This is easily shown from equation (21.6) as
† −1†ia ia Tˆˆ( a) = T( −=+ a) 11 pˆˆ =− p = Taˆ( ) , (21.7)
2 where in the last steps use has been made of the Hermitian property of the momentum operator.
Transformations of operators Consider an operator Qˆ. If we translate a wave function, then the expectation value of Qˆ will be
ψψ′′Qˆˆ= Tˆ ψ Q T ˆ ψψ= T ˆˆ† QT ˆ ψ. (21.8)
We see that the expectation value of Qˆ using the translated wave function is the same as the expectation value of Qˆ′ using the untranslated wave function, where
Qˆˆ′ = Tˆˆ† QT. (21.9)
Now, we see that we have a choice. We can either transform wave functions keeping operators unchanged or we can keep the original wave functions but transform the operators as in equation (21.9). This duality is nicely illustrated by example 6.1 in GS. Consider the operator xˆ. Applying the translated operator to an arbitrary function f(x), we get
xfxˆˆ′ ( ) = Tˆˆ†( axTafx) ( ) ( ) = T ˆ−− 11( axfxa) ˆ( −=) T ˆ( axfxa) ( −=+) ( xafx) ( ). (21.10)
We see that, because f(x) is arbitrary,
xˆˆ′ = xa + . (21.11)
This corresponds to shifting the coordinate origin to the left by a, which is equivalent to keeping the origin fixed but moving functions to the right by a.
Because it involves a derivative with respect to position, the momentum operator is invariant under translation. It can then be shown that an operator that depends on xˆ and pˆ transforms under translation as
Q′( xpˆˆ,) = Qx( ˆ + ap ,. ˆ) (21.12)
Translational symmetry A system is translationally invariant if its Hamiltonian is unchanged by the transformation, i.e.
Hˆ′ = T ˆ† HT ˆˆ = H ˆ. (21.13)
Since the translation operator is unitary, this condition can be written as
ˆˆ ˆˆ ˆ ˆ HT=⇒= TH H, T 0. (21.14)
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The system is translationally invariant if its Hamiltonian commutes with the translation operator. Since translation does not change the momentum operator, the Hamiltonian for a single particle will be translationally invariant if
Vx( += a) Vx( ). (21.15)
If this is true for all a, there is a continuous symmetry in which V is constant. For a discrete symmetry, the condition is true only for a discrete set of values of a.
Bloch’s Theorem Suppose the potential energy has a discrete translational symmetry, as in a crystal structure. Since ˆˆ ˆ ˆ ˆ HT,= 0, H and T have a common set of eigenfunctions. Since T is unitary, its eigenvalues have absolute value 1. By convention, the eigenvalues of Tˆ are written as
λ = e−iqa , (21.16)
where q is called the crystal momentum. Applying Tˆ to a stationary state eigenfunction, we get
Taˆ ( )ψ( x) = λψ( x) ⇒ ψ ( x −= a) e−iqaψ ( x). (21.17)
Now let
ψ ( x) = eiqx ux( ). (21.18)
Then
iq( x− a) − e ux( −= a) eiqa e iqx ux( ) ⇒ ux( −= a) ux( ). (21.19)
We see that the stationary state wave function is a travelling wave (i.e. the wave function for a free particle) multiplied by a periodic function of x. This is Bloch’s Theorem. The travelling wave has a non- zero velocity. This means an electron could move through a perfect crystal without resistance.
Translational symmetry and momentum conservation If the Hamiltonian has a continuous translational symmetry, then for any infinitesimal translation
ia Taˆ ( ) =1. − pˆ (21.20)
Since the Hamiltonian and translation operators commute, we see that the Hamiltonian commutes with the momentum operator. From the generalized Ehrenfest theorem
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di∂ Qˆ Q= HQˆ ,,ˆ + (21.21) dt ∂t we see that the expectation value of the momentum is conserved.
Conservation Laws More generally, from equation (21.21), we see that the expectation value of any time-independent transformation that leaves the Hamiltonian invariant will be a conserved quantity. A more stringent statement of a conservation law is that the probability of measuring any particular value of an observable is independent of time. To see why, consider a system with a time-independent Hamiltonian. The wave function at any time is then
−ωnt Ψ=(t) ∑ cennψ , (21.22) n
where the cn’s are a set of constant that depend on the initial state. Now consider an observable, Q. If
ˆ ˆ = then the ψ are also eigenfunctions of ˆ Let qm be an eigenvalue of ˆ Then the HQ, 0, n Q. Q. probability of getting qm on measurement of Q is
2 2 2 −itωn p( qm) =Ψ=ψ m (t) ∑ cenψψ mn=c m, (21.23) n
which is time independent.
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