12.5 The propagator in position space

Concept links · terms present in this machine draft; source roles are unverified: eigenvalue

Free particle states with the simplest physical interpretation are momentum eigenstates. They describe a single quantum particle with a fixed momentum $k^{'}$ , and this momentum is a conserved quantity that will not change. In the momentum space representation (see section 11.5) such a time-dependent state will be given by

ψ (k, t) = e^{- i \frac{1}{2 m} k^{'2} t} δ (k - k^{'})

In the position space representation such a state will be given by

ψ (q, t) = \frac{1}{2 π} e^{- i \frac{1}{2 m} k^{'2} t} e^{i k^{'} q}

a wave with (restoring temporarily factors of ℏ and using $p = ℏ k)$ wavelength $\frac{2 π ℏ}{p ^{'}}$ and angular frequency $\frac{p ^{'} ^{2}}{2 m ℏ}$

As for any quantum system, time evolution of a free particle from time 0 to time $t$ is given by a unitary operator $U (t) = e^{- i t H}$ . In the momentum space representation this is just the multiplication operator

U (t) = e^{- i \frac{1}{2 m} k^{2} t}

In the position space representation it is given by an integral kernel called the “propagator”:

Definition (Position space propagator)

The position space propagator is the kernel $U (t, q_{t}, q_{0})$ of the time evolution operator acting on position space wavefunctions. It determines the time evolution of wavefunctions for all times t $b y$

$ψ (q_{t}, t) = \int_{- \infty}^{+ \infty} U (t, q_{t}, q_{0}) ψ (q_{0}, 0) d q_{0} (12.4)$
where $ψ (q_{0}, 0)$ is the initial value of the wavefunction at time $0$ .

In the Dirac notation one has

ψ (q_{t}, t) = ⟨ q_{t} ∣ ψ (t)⟩ = ⟨ q_{t} ∣ e^{- i H t} ∣ ψ (0)⟩ = ⟨ q_{t} ∣ e^{- i H t} \int_{- \infty}^{\infty} ∣ q_{0} ⟩ ⟨ q_{0} ∣ ψ (0)⟩ d q_{0}

and the propagator can be written as

U (t, q_{t}, q_{0}) = ⟨ q_{t} ∣ e^{- i H t} ∣ q_{0} ⟩

$U (t, q_{t}, q_{0})$ can be computed for the free particle case by Fourier transform of the momentum space multiplication operator:

ψ (q_{t}, t) = \frac{1}{2 π} \int_{- \infty}^{+ \infty} e^{ik q_{t}} ψ (k, t) d k = \frac{1}{2 π} \int_{- \infty}^{+ \infty} e^{ik q_{t}} e^{- i \frac{1}{2 m} k^{2} t} ψ (k, 0) d k = \frac{1}{2 π} \int_{- \infty}^{+ \infty} e^{ik q_{t}} e^{- i \frac{1}{2 m} k^{2} t} (\int_{- \infty}^{+ \infty} e^{- ik q_{0}} ψ (q_{0}, 0) d q_{0}) d k = \int_{- \infty}^{+ \infty} (\frac{1}{2 π} \int_{- \infty}^{+ \infty} e^{ik (q_{t} - q_{0})} e^{- i \frac{1}{2 m} k^{2} t} d k) ψ (q_{0}, 0) d q_{0}

U (t, q_{t}, q_{0}) = U (t, q_{t} - q_{0}) = \frac{1}{2 π} \int_{- \infty}^{+ \infty} e^{ik (q_{t} - q_{0})} e^{- i \frac{1}{2 m} k^{2} t} d k (12.5)

Note that (as expected due to translation invariance of the Hamiltonian operator) this only depends on the diference $q_{t} - q_{0}$ . Equation 12.5 can be rewritten as an inverse Fourier transform with respect to this diference

U (t, q_{t} - q_{0}) = \frac{1}{2 π} \int_{- \infty}^{+ \infty} e^{ik (q_{t} - q_{0})} U (t, k) d k

where

U (t, k) = \frac{1}{2 π} e^{- i \frac{1}{2 m} k^{2} t} (12.6)

To make sense of the integral 12.5, the product it can be replaced by a complex variable $z = τ + i t$ . The integral becomes well-defined when $τ = Re (z)$ (“imaginary time”) is positive, and then defines a holomorphic function in $z$ . Doing the integral by the same method as in equation 11.6, one finds

U (z = τ + i t, q_{t} - q_{0}) = \frac{m}{2 π z} e^{- \frac{m}{2 z} (q_{t} - q_{0})^{2}} (12.7)

For $z = τ$ real and positive, this is the kernel function for solutions to the partial diferential equation

\frac{\partial}{\partial τ} ψ (q, τ) = \frac{1}{2 m} \frac{\partial ^{2}}{\partial q ^{2}} ψ (q, τ)

known as the “heat equation”. This equation models the way temperature difuses in a medium, it also models the way probability of a given position difuses in a random walk. Note that here it is $ψ (q)$ that gives the probability density, something quite diferent from the way probability occurs in measurement theory for the free particle quantum system. There it is $∣ ψ ∣^{2}$ that gives the probability density for the particle to have position observable eigenvalue q.

Taking as initial condition

ψ (q_{0}, 0) = δ (q_{0} - q^{'})

the heat equation will have as solution at later times

ψ (q_{τ}, τ) = \frac{m}{2 π τ} e^{- \frac{m}{2 τ} (q^{'} - q_{τ})^{2}} (12.8)

This is physically reasonable: at times $τ > 0.$ , an initial source of heat localized at a point $q^{'}$ difuses as a Gaussian about $q^{'}$ with increasing width. For $τ < 0,$ one gets something that grows exponentially at ±∞, and so is not in $L^{2} (R)$ or even $S^{'} (R)$ .

In real time t as opposed to imaginary time $τ (i.e., z = i t$ , interpreted as the limit lim $_{ϵ 0^{+}} (ϵ + i t))$ ), equation 12.7 becomes

U (t, q_{t} - q_{0}) = \frac{m}{i 2 π t} e^{- \frac{m}{i 2 t} (q_{t} - q_{0})^{2}} (12.9)

Unlike the case of imaginary time, this expression needs to be interpreted as a distribution, and as such equation 12.4 makes sense for $ψ (q_{0}, 0) \in S (R)$ . One can show that, for $ψ (q_{0}, 0)$ with amplitude peaked around a position $q^{'}$ and with amplitude of its Fourier transform peaked around a momentum $k^{'}$ , at later times $ψ (q, t)$ will become less localized, but with a maximum amplitude at $q^{'} + \frac{k ^{'}}{m} t$

Figure 12.1: Time evolution of an initially localized wavefunction.

This is what one expects physically, since $\frac{p ^{'}}{m}$ is the velocity corresponding to momentum $p^{'}$ for a classical particle.

Note that the choice of square root of $i$ in 12.9 is determined by the condition that one get an analytic continuation from the imaginary time version for $τ > 0$ so one should take in 12.9

\frac{m}{i 2 π t} = e^{- i \frac{π}{4}} \frac{m}{2 π t}

We have seen that an initial momentum eigenstate

ψ (q_{0}, 0) = \frac{1}{2 π} e^{i k^{'} q_{0}}

evolves in time by multiplication by a phase factor. An initial position eigenstate

ψ (q_{0}, 0) = δ (q_{0} - q^{'})

evolves to

ψ (q_{t}, t) = \int_{- \infty}^{+ \infty} U (t, q_{t} - q_{0}) δ (q_{0} - q^{'}) d q_{0} = \frac{m}{i 2 π t} e^{- \frac{m}{i 2 t} (q_{t} - q^{'})^{2}}

Near $t = 0$ this function has a rather peculiar behavior. It starts out localized at $q_{0}$ at $t = 0$ , but at any later time $t > 0,$ , no matter how small, the wavefunction will have constant amplitude extending out to infinity in position space. Here one sees clearly the necessity of interpreting such a wavefunction as a distribution.

For a physical interpretation of this calculation, note that while a momentum eigenstate is a good approximation to a stable state one can create and then study, an approximate position eigenstate is quite diferent. Its creation requires an interaction with an apparatus that exchanges a very large momentum (involving a very short wavelength to resolve the position). By the Heisenberg uncertainty principle, a precisely known position corresponds to a completely unknown momentum, which may be arbitrarily large. Such arbitrarily large momenta imply arbitrarily large velocities, reaching arbitrarily far away in arbitrarily short time periods. In later chapters we will see how relativistic quantum theories provide a more physically realistic description of what happens when one attempts to localize a quantum particle, with quite diferent phenomena (including possible particle production) coming into play.

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12.5 The propagator in position space

12.5 The propagator in position space

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