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3.5: Propagators and Representations

 

Introduction

We’ve spent most of the course so far concentrating on the eigenstates of the Hamiltonian, states whose time-dependence is merely a changing phase.  We did mention much earlier a superposition of two different energy states in an infinite well, resulting in a wave function sloshing backwards and forwards. It’s now time to cast the analysis of time dependent states into the language of bras, kets and operators.  We’ll take a time-independent Hamiltonian H,<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> with a complete set of orthonormalized eigenstates, and as usual

iψ(x,t)t=22m2ψ(x,t)x2+V(x)ψ(x,t),<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>

Or, as we would now write it

it|ψ(x,t)=H|ψ(x,t).<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>

Since H<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> is itself time independent, this is very easy to integrate!

|ψ(x,t)=eiH(tt0)/|ψ(x,t0).<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>

The exponential operator that generates the time-dependence is called the propagator, because it describes how the wave propagates from its initial configuration, and is usually denoted by U:<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>  

 

|ψ(x,t)=U(tt0)|ψ(x,t0).<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>

It’s appropriate to call the propagator U<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>, because it’s a unitary operator:

U(tt0)=eiH(tt0), so U(tt0)=eiH(tt0)=eiH(tt0)=U1(tt0).<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>

Since H<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> is Hermitian, U<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> is unitary.  It immediately follows that

ψ(x,t)|ψ(x,t)=ψ(x,t0)|UU(tt0)|ψ(x,t0)=ψ(x,t0)|ψ(x,t0)<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>

the norm of the ket vector is conserved, or, translating to wave function language, a wave function correctly normalized to give a total probability of one stays that way.  (This can also be proved from the Schrödinger equation, of course, but this is quicker.)

 

This is all very succinct, but unfortunately the exponential of a second-order differential operator doesn’t sound too easy to work with.  Recall, though, that any function of a Hermitian operator has the same set of eigenstates as the original operator.  This means that the eigenstates of eiH(tt0)/<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> are the same as the eigenstates of H<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>, and if H|ψn=En|ψn,<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> then

eiH(tt0)/|ψn=eiEn(tt0)/|ψn.<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>

 

This is of course nothing but the time dependent phase factor for the eigenstates we found before<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>and, as before, to find the time dependence of any general state we must express it as a superposition of these eigenkets, each having its own time dependence.  But how do we do that in the operator language?  Easy: we simply insert an identity operator, the one constructed from the complete set of eigenkets, thus:

|ψ(t)=eiH(tt0)/n=1|ψnψn|ψ(t0)=n=1eiEn(tt0)/|ψnψn|ψ(t0).<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>

Staring at this, we see that it’s just what we had before: at the initial time t=t0,<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> the wave function can be written as a sum over the eigenkets:

 

|ψ(t0)=|ψn(t0)ψn(t0)|ψ(t0)=cn|ψn(t0)<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>

 

with  cn=ψn|ψ<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>,  |cn|2=1<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>, and the usual generalization for continuum eigenvalues, and the time development is just given by inserting the phases:

|ψ(t)=cneiEn(tt0)/|ψn(t0).<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>

The expectation value of the energy E<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> in |ψ<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>,

E=ψ|H|ψ=|cn|2En<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>

and is (of course) time independent.

 

The expectation value of the particle position x is

ψ(t)|x|ψ(t)=n,mcncmei(EnEm)(tt0)/ψn(t0)|x|ψm(t0)<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>

and is not in general time-independent.  (It is real, of course, on adding the n,m<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> term to the m,n<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> term.)

 

This analysis is only valid for a time-independent Hamiltonian. The important extension to a system in a time-dependent external field, such as an atom in a light beam, will be given later in the course.

The Free Particle Propagator

To gain some insight into what the propagator U<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> looks like, we’ll first analyze the case of a particle in one dimension with no potential at all.  We’ll also take t0=0<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> to make the equations less cumbersome.

 

For a free particle in one dimension E=p2/2m=2k2/2m<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> the energy eigenstates are also momentum eigenstates, we label them |k<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>, so

 

U(t)=eiHt/=eiHt/dk2π|kk|=eik2t/2mdk2π|kk|.<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>, a particle is at x0<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>:  ψ(x,t=0)=δ(xx0)=|x0<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>: what is the probability amplitude for finding it at x<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> at a later time t<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>?  (This would be just its wave function at the later time.)

x|U(t,0)|x0=eik2t/2mdk2πx|kk|x0=eik2t/2mdk2πeik(x0x)=m2πiteim(x0x)2/2t,<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>

On examining the above expression, though, it turns out to be nonsense!  Noting that the term in the exponent is pure imaginary, |ψ(x,t)|2=m/2πt<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> independent of x<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>!  This particle apparently instantaneously fills all of space, but then its probability dies away as 1/t<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> 

 

Question: Where did we go wrong?

 

Answer:  Notice first that |ψ(x,t)|2<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> is constant throughout space.  This means that the normalization, |ψ(x,t)|2dx=<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>!  And, as we’ve seen above, the normalization stays constant in time<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>the propagator is unitary.  Therefore, our initial wave function must have had infinite norm.  That’s exactly right<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>we took the initial wave function ψ(x,t=0)=δ(xx0)=|x0<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>.

Think of the δ<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> -function as a limit of a function equal to 1/Δ<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> over an interval of length Δ<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>, with Δ<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> going to zero, and it’s clear the normalization goes to infinity as 1/Δ<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>.  This is not a meaningful wave function for a particle.  Recall that continuum kets like|x0<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> are normalized by x|x=δ(xx)<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>, they do not represent wave functions individually normalizable in the usual sense.  The only meaningful wave functions are integrals over a range of such kets, such as dxψ(x)|x<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>.  In an integral like this, notice that states |x<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> within some tiny x<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> -interval of length δx,<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> say, have total weight ψ(x)δx<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>, which goes to zero asδx<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> is made smaller, but by writing ψ(x,t=0)=δ(xx0)=|x0<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> we took a single such state and gave it a finite weight.  This we can’t do.

 

Of course, we do want to know how a wave function initially localized near a point develops.  To find out, we must apply the propagator to a legitimate wave function<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>one that is normalizable to begin with. The simplest “localized particle” wave function from a practical point of view is a Gaussian wave packet,

ψ(x,0)=eip0x/ex2/2d2(πd2)1/4.<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>

(I’ve used d<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> in place of Shankar’s Δ<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> here to try to minimize confusion with Δx,<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> etc.)

 

The wave function at a later time is then given by the operation of the propagator on this initial wave function:  

ψ(x,t)=U(x,t;x,0)eip0x/ex2/2d2(πd2)1/4dx=m2πiteim(xx)2/2teip0x/ex2/2d2(πd2)1/4dx.<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>

The integral over x<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> is just another Gaussian integral, so we use the same result,

 

dxeax2+bx=πaeb2/4a<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>.

 

Looking at the expression above, we can see that

b=imt(xp0tm)<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>,   a=12d2im2t<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>.

This gives

ψ(x,t)=π1/4d(1+itmd2)exp(imx22t)exp(imt(xp0tm)22(1+itmd2))<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>

where the second exponential is the term eb2/4a<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>.  As written, the small t<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> limit is not very apparent, but some algebraic rearrangement yields:

ψ(x,t)=π1/4d(1+it/md2)exp((xp0t/m)22d2(1+it/md2))exp(ip0(xp0t/2m))<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>.

 

It is clear that this expression goes to the initial wave packet as t<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> goes to zero.  Although the phase has contributions from all three terms here, the main phase oscillation is in the third term, and one can see the phase velocity is one-half the group velocity, as discussed earlier.

 

The resulting probability density:

|ψ(x,t)|2=1π(d2+2t2/m2d2)exp(xp0t/m)2(d2+2t2/m2d2)<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>.

This is a Gaussian wave packet, having a width which goes as t/md<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> for large times, where d<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> is the width of the initial packet in x<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> -space<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>so /md<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> is the spread in velocities Δv<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> within the packet, hence the gradual spreading  Δvt<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> in x<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> -space.

 

It’s amusing to look at the limit of this as the width d<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> of the initial Gaussian packet goes to zero, and see how that relates to ourδ<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> -function result.  Suppose we are at distance x<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> from the origin, and there is initially a Gaussian wave packet centered at the origin, width dx.<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> At time tmxd/<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>, the wave packet has spread to x<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> and has |ψ(x,t)|2<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> of order 1/x<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> at x.<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> Thereafter, it continues to spread at a linear rate in time, so locally |ψ(x,t)|2<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> must decrease as 1/t<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> to conserve probability.  In the δ<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> -function limit d0<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>, the wave function instantly spreads through a huge volume, but then goes as 1/t<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> as it spreads into an even huger volume.  Or something.

Schrödinger and Heisenberg Representations

Assuming a Hamiltonian with no explicit time dependence, the time-dependent Schrödinger equation has the form

it|ψ(x,t)=H|ψ(x,t)<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>

and as discussed above, the formal solution can be expressed as:

|ψ(x,t)=eiHt/|ψ(x,t=0).<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>

Now, any measurement on a system amounts to measuring a matrix element of an operator between two states (or, more generally, a function of such matrix elements). 

 

In other words, the physically significant time dependent quantities are of the form

φ(t)|A|ψ(t)=φ(0)|eiHt/AeiHt/|ψ(0)<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>

where A<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> is an operator, which we are assuming has no explicit time dependence.

 

So in this Schrödinger picture, the time dependence of the measured value of an operator like x<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> or p<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> comes about because we measure the matrix element of an unchanging operator between bras and kets that are changing in time.

 

Heisenberg took a different approach: he assumed that the ket describing a quantum system did not change in time, it remained at |ψ(0),<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> but the operators evolved according to:

AH(t)=eiHt/AH(0)eiHt/.<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>

 

Clearly, this leads to the same physics as before. The equation of motion of the operator is:

idAH(t)dt=[AH(t),H].<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>

The Hamiltonian itself does not change in time<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>energy is conserved, or, to put it another way, H<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> commutes with eiHt/.<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>  But for a nontrivial Hamiltonian, say for a particle in one dimension in a potential,

H=p2/2m+V(x)<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>

 

the separate components will have time-dependence, parallel to the classical case: the kinetic energy of a swinging pendulum varies with time.  (For a particle in a potential in an energy eigenstate the expectation value of the kinetic energy is constant, but this is not the case for any other state, that is, for a superposition of different eigenstates.)  Nevertheless, the commutatorof x,p<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> will be time-independent:

[xH(t),pH(t)]=eiHt/[xH(0),pH(0)]eiHt/=eiHt/ieiHt/=i.<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>

(The Heisenberg operators are identical to the Schrödinger operators at t=0.<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> ) 

Applying the general commutator result [A,BC]=[A,B]C+B[A,C]<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>,

[xH(t),p2H(t)2m]=ipH(t)m<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>

so

dxH(t)dt=pH(t)m<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>

 

and since [xH(t),pH(t)]=i,  pH(t)=id/dxH(t)<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>,

 

dpH(t)dt=1i[pH(t),V(xH(t))]=V(xH(t)).<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>

 

This result could also be derived by writing V(x)<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> as an expansion in powers of x,<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> then taking the commutator with p.<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>

 

Exercise: check this.

 

Notice from the above equations that the operators in the Heisenberg Representation obey the classical laws of motion! Ehrenfest’s Theorem, that the expectation values of operators in a quantum state follow the classical laws of motion, follows immediately, by taking the expectation value of both sides of the operator equation of motion in a quantum state.

Simple Harmonic Oscillator in the Heisenberg Representation

For the simple harmonic oscillator, the equations are easily integrated to give:

xH(t)=xH(0)cosωt+(pH(0)/mω)sinωtpH(t)=pH(0)cosωtmωxH(0)sinωt.<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>

We have put in the H<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> subscript to emphasize that these are operators.  It is usually clear from the context that the Heisenberg representation is being used, and this subscript  may be safely omitted.

 

The time-dependence of the annihilation operator a<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> is:

a(t)=eiHt/a(0)eiHt/<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>

with

H=ω(a(t)a(t)+12/).<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>

 

Note again that although H<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> is itself time-independent, it is necessary to include the time-dependence of individual operators within H.<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>  

iddta(t)=[a(t),H]=ω[a(t),a(t)a(t)]=ω[a(t),a(t)]a(t)=ω  a(t)<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>

so

a(t)=a(0)eiωt.<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>

Actually, we could have seen this as follows: if |n<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> are the energy eigenstates of the simple harmonic oscillator,

eiHt/|n=einωt/|n=einωt|n.<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>

Now the only nonzero matrix elements of the annihilation operator aˆ<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> between energy eigenstates are of the form

 

n1|a(t)|n=n1|eiHt/a(0)eiHt/|n=eiω(n1)tn1|a(0)|neiωnt=n1|a(0)|neiωt.<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>

 

Since this time-dependence is true of all energy matrix elements (trivially so for most of them, since they’re identically zero), and the eigenstates of the Hamiltonian span the space, it is true as an operator equation.

 

Evidently, the expectation value of the operator a(t)<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> in any state goes clockwise in a circle centered at the origin in the complex plane. That this is indeed the classical motion of the simple harmonic oscillator is confirmed by recalling the definition a=ξ+iπ2=12mω(mωx+ip)<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math>, so the complex plane corresponds to the (mωx,p)<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics/></math> phase space discussed near the beginning of the lecture on the Simple Harmonic Oscillator. We’ll discuss this in much more detail in the next lecture, on Coherent States.

 

The time-dependence of the creation operator is just the adjoint equation: a(t)=a(0)eiωt.