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The lowestenergyeigenvalueis the ground state energyandthe correspond-
ingeigenstateissimplytheground state. Itisinterestingtoinserttheexpansion
(5.60) into the Schr¨odinger equation. A short calculation yields the following
equation holding for an arbitrary energy state
dn
i¯h | i =E n ,
n
dt | i
the solution of which is
n =exp( iE t/¯h)n .
n
| i − | i
So,inacertainsense,thetimedependenceforenergyeigenstatesaretrivial,
itis justanoveralloscillatingphasefactor. Itdoesplayaroleinsuperpositions
though, where different eigenstates oscillates with different frequencies. The
frequency of oscillation ω is defined by ω = E/¯h so that the energy is often
written as E =¯hω .
n n
Unitary transformation
The time development of a quantum system between two times t and t can
1 2
also be described by a unitary transformation. Let ψ(t ) and ψ(t ) be the
1 2
| i | i
state at the two times respectively. Then
ψ(t ) =U(t ,t )ψ(t ) (5.61)
2 1 2 1
| i | i
whereU(t ,t )is a unitary operatorthatdepends only onthe twotimes t and
1 2 1
t , i.e. there is no other time dependence in U.
2
The two ways of prescribing the dynamics of the quantum state can be
related by formally solving the Schr¨odinger equation. Provided we grant us
the privilege to formally exponentiate the Hamiltonian operator, a solution to
equation (5.59) can be written
ψ(t) =exp( iHt/¯h)ψ(0) . (5.62)
| i − | i
The intuition here is that the state starts in the state ψ(0) at an initial
| i
time t = 0 and develops into ψ(t) at time t. Let us check this by a short
| i
calculation
d d
i¯h ψ(t) = i¯h exp( iHt/¯h) ψ(0) =
dt| i dt − | i
(cid:0) (cid:1)
(i¯h)( iH/¯h)exp( iHt/¯h)ψ(0) =H ψ(t)
− − | i | i
115
Clearly, this calculation presupposes that the Hamiltonian has no explicit time
dependence.
Equation (5.62) can also be written somewhat more generally as
ψ(t ) =exp( iH(t t )/¯h)ψ(t ) , (5.63)
2 2 1 1
| i − − | i
expressing time development from time t to time t . Now comparing this
1 2
formal solution to the Schr¨odinger equation we see the connection between the
two ways of prescribing the dynamics of the system. The unitary operator U
should be equated to the exponential of the Hamiltonian, or
U(t ,t )=exp( iH(t t )/¯h).
1 2 2 1
− −
Unitarity and reversibility
The unitarity of time evolution leads immediately to the reversibility of the
dynamics. Since U(t ,t ) is invertible we can recover the ’initial’ state ψ(t )
1 2 1
| i
from the ’final’ state ψ(t ) by multiplying the equation (5.61) by U(t ,t )
2 1 2 †
| i
U(t 1,t 2)† ψ(t 2) =U(t 1,t 2)†U(t 1,t 2)ψ(t 1) = ψ(t 1) .
| i | i | i
There is one more issue that must be clarified in the context of quantum
dynamics, and that is the different so called ’pictures’. In quantum dynamics,
there is a choice as to where the time dependence resides. One choice is to
let the states carry all the time dependence and letting the operators be time
independent. This is the Schro¨dinger picture. Another choice is to have the
statesthemselvesbetimeindependentandlettingalltimedependencebecarried
by the operators. This is the Heisenberg picture. There are also intermediate
pictures,wherethetimedependenceissplitinawelldefinedwaybetweenstates
and operators. One such picture is the interaction picture, which is useful in
calculations.
5.5.1 Schr¨odinger picture
Of the Schr¨odinger picture there is not much more to be said. In fact, it is
the Schr¨odingerpicture that we haveimplicitly usedin the preceeding sections.
The Hamiltonian and all other operators are time independent and the time
dependence is carriedby the states. Formally solving the Schr¨odinger equation
as in (5.62) makes this explicit.
5.5.2 Heisenberg picture
The transition to the Heisenberg picture is interesting and we will carry it
through in some detail. First let ψ (t) and φ (t) be two quantum states
S S
| i | i
where the index S is used to indicate that these are taken in the Schr¨odinger
picture. Laterin the discussionwe will introduce the correspondingHeisenberg
picture states ψ = ψ (0) and φ = φ (0) . Next consider an operator
H S H S
| i | i | i | i