Theoretical background
In classical mechanics the description of the dynamic state of a particle at a given time t is based on the specification of six parameters, the components of the position r(t) and linear momentum p(t) of the particle. All the dynamical variables (energy, angular momentum, etc.) are determined by the specification of r(t) and p(t). Newton's laws enable us to calculate r(t) through the solution of second order differential equations with respect to time. Consequently, they fix the values of r(t) and p(t) for any time t when they are known for the initial time.
Quantum mechanics uses a more complicated description of phenomena. The dynamic state of a particle, at a given time, is characterised by a wave function Psi(r,t) which contains all the information that is possible to obtain about the particle. The state no longer depends on six parameters, but on an infinite number of parameters: the values of the wave function Psi(r,t) at all points r in the coordinate space. For the classical idea of trajectory (the succession in time of the various states of the classical particle) we must substitute the idea of the propagation of the wave associated with the particle. Psi(r,t) is interpreted as the probability amplitude of the particles presence. Abs[Psi(r,t)]2 is interpreted as the probability density of the particle being, at time t, in a volume element d3r situated at the point r. The equation describing the evolution of the wave function Psi(r,t) is the Schrödinger equation. The result of a measurement of an arbitrary dynamic variable must belong to the set of the eigen values of the operator representing the dynamic variable. With each eigen value it is associated an eigenstate, the eigen function of the operator belonging to the particular eigen value. If a measurement yields a particular eigen value, the corresponding eigen function is the wave function of the particle immediately after the measurement. The predictions of the measurement results are only probabilistic: they yield the probability of obtaining a given result in the measurement of a dynamical variable.
When a particle of mass m is subjected to the influence of a scalar potential V(r,t), the Hamiltonian operator of the particle in Schrödinger representation can be expressed as H = T + V(r,t) , where T is the kinetic energy operator of the form
The Schrödinger equation
,
(1)
which governs the time evolution of the physical system is of first order in t. From this it follows that, given the initial state Psi(r,t)0, the final state Psi(r,t) at any subsequent time t is determined. There is no indeterminacy in the time evolution of a quantum system. Indeterminacy appears only when a physical quantity is measured, the state function then undergoing an unpredictable modification. However, between two measurements, the state function evolves in a perfectly deterministic way in accordance with equation (1). The Schrödinger equation is linear and homogenous; its solutions are linearly superposable which leads to wave effects.
Consider a particle whose potential energy is zero (or has a constant value) at every point in space. The particle is not subjected to any force; it is said to be free. When V(r,t) = 0, the Schrödinger equation becomes:
(2)
and a plane wave of type
,
(3)
(where A is a constant) is a solution of it, on the condition that k and Omega satisfy the dispersion relation for a free particle:
.
(4)
Since Abs[Psi(r,t)]2 = Abs[A]2, a plane wave of this type represents a particle whose probability of presence is uniform throughout all space. From the principle of superposition it follows that every linear combination of plane waves satisfying (4) will also be a solution of the Schrödinger equation. Such a superposition can be written as
.
(5)
d3k represents, by definition, the infinitesimal volume element in the k-space. g(k), which can be complex, must be sufficiently regular to allow differentiation inside of the integral. A wave function, such as (5), a superposition of plane waves, is called a three dimensional wave packet.
A plane wave, whose modulus is constant throughout all space, is not square-integrable, therefore, rigorously, it cannot represent a physical state of the particle. On the other hand, the superposition of plane waves can be square-integrable. It can be shown that any square-integrable solution can be written in the form (5). The form of the wave packet at a given instant of time, if we choose this instant as the time origin, is :
.
(6)
It can be seen that g(k) is the Fourier transform of Psi(r,0):
.
(7)
Consequently the validity of formula (6) is not limited to the case of the free particle: whatever the potential, Psi(r,0) can always be written in this form. In the general case, where the potential V(r) is arbitrary, the formula (4) is not valid. It is then useful to introduce the three dimensional Fourier transform g(k,t) of the function Psi(r,t) by writing:
.
(8)
The time dependence of g(k,t) is brought in and determined by the potential V(r). The wave function's representation in the coordinate space, Psi(r,t) and its representation g(k,t) in the k space, form a Fourier transform pair:
.
(9)
The velocity of propagation of the wave packet (of the envelope of the wave packet) is the group velocity:
.
(10)
Consider, in a one dimensional model, a free particle [V(x) 0]. A normalised gaussian wave packet can be obtained by superposing plane waves eikx with the coefficients:
.
(11)
Expression (11) corresponds to a gaussian function centred at k = k0 (and multiplied by a numerical coefficient which normalises the wave function). The wave function at time t = 0 is:
,
(12)
which shows that the Fourier transform of a gaussian function is also gaussian.Therefore at time t = 0, the probability density of the particle is given by
. (13)
The centre of the wave packet is at x=0. It is convenient to define the width of the wave function by Delta x, the root-mean-square deviation of the x, thus Delta x = a/2. Since Abs[g(k,0)]2 is also a gaussian function we can calculate its width in a similar way; this gives Deltak = 1/a.
The wave function of the free particle Psi(x,t) at time t is
(14)
with
dispersion relation. By performing the integral in expression (14) it can be shown that at any time t the envelop of the wave packet remains gaussian but it is spreading out in time. The width of the wave packet (Delta x) is a function of time
.
(15)
The height of the wave packet also varies, it decreases as the wave packet is spreading out, so the norm of Psi(x,t) remains constant.
The properties of the function g(k,t), the Fourier transform of Psi(x,t), are completely different. From equation (14) it follows that
.
(16)
g(k,t) has the same norm as g(k,0)
therefore the average momentum
of the wave packet and its momentum dispersion
do not vary in time. This arises from the fact that the momentum is a constant
of motion for a free particle; since the particle encounters no obstacle, its
momentum distribution cannot change.
To follow the evolution of state of the system one has to solve the quantum-mechanical equation of motion - the time-dependent Schrödinger equation. Analytical solution exists only for some oversimplified cases.
In one dimension one can find the solution for an arbitrary V(x)
potential by numerical integration of the time-dependent Schrödinger
equation. This is performed such, that first the effect of the Hamiltonian on
Psi(x,t)0, the state function at the initial time
instant initial t0, is calculated. This gives the time rate
of change of the state function at the initial time instant
t0. From this one gets the change of the state function for
the time interval
, and thus the state function at the time instant t,
Psi(x,t). By choosing short time intervals and close values of the
x coordinate the method provides a good approximation of the evolution
of the state function.
To get the evolution of the wave function by the outlined method in two or three dimensions is rather hopeless, mainly because of computational time limitations. These limitations are largely removed by an efficient numerical technique based on splitting of the evolution operator and Fourier transform technique.
The transformation of Psi(r,t)0 (the state function at an initial instant t0) into Psi(r,t) (the state function at an arbitrary instant t) is linear. Therefore there exists a linear operator U(t, t0), such that:
.
(17)
The operator U(t,t0) is, by definition, the evolution operator of the system. U(t,t0) is a unitary operator, it conserves the norm of vectors on which it acts. Unitarity expresses the conservation of probability. In case of conservative systems, when the operator H does not depend on time, equation (17) can easily be integrated and we obtain:
,
(18)
where
.
Since the Hamiltonian operator H contains the scalar function V(r) and through T the differential operator, it is difficult to evaluate U(t,t0) in the numerical solution. The evaluation of the exponential by expressing it as a power series and truncating that would lead to the loss of unitarity of the evolution operator, that is to non-conservation of probability and should be avoided. The splitting of the evolution operator into the product of two exponentials: one containing only derivative operators, the other the scalar function V(r) is more rewarding, because if both terms can be evaluated exactly it preserves unitarity. But because the kinetic energy operator T and the potential energy operator V(r) do not commute, the splitting of the evolution operator into the product of two exponentials
(19)
is only an approximation. According to Glauber's formula, if two operators A and B commute with their commutator [A,B] the relation
(20)
is valid, and the error introduced in approximating the evolution operator by
(19) is of
.
The approximation can be done considerable better by a slight modification to
it by decomposing the evolution operator symmetrically:
(21)
which leads to the reduction of the error to
.
Note, that if the potential V is constant, T and V commute
and thus the error introduced by the splitting of the evolution operator into
the product of exponentials vanishes. This means that equation (21) treats the
motion of a free particle exactly.
According to equation (21) the evaluation of the action of the evolution
operator on the Psi(r,t) wavefunction is split into three consecutive
steps. The potential energy operator of the system is a scalar function in
coordinate space therefore the evaluation of the effect of the operator
on the wave function is only a multiplication by it. Evaluation of the effect
of the operator
is more difficult, as the kinetic energy operator T contains the
differential operator in coordinate space. To evaluate its action on the wave
function one can utilise the property of the Fourier transform that the act of
differentiation on a function in coordinate space is equivalent to the act of
multiplication on that function's representation in the Fourier transform space
by k, the Fourier transform variable conjugate to the coordinate. This
means that the kinetic energy operator T is only a function of the wave
vector k in the momentum space, T =
hBar2k2/2m. Thus the action of the
exponential containing the kinetic energy operator on the wavefunction can be
evaluated as
,
(22)
where the inverse Fourier transform is denoted by F-1.
The evolution of the wave function over a time increment is calculated in a
straightforward way: first equation (22) is applied, then the result is
multiplied by
and finally equation (22) is applied again. Thus the exact evolution is
approximated by the product of a free particle evolution for one-half the time
increment, a potential only evolution for a full time increment, and a final
free particle evolution for another half time increment. Convergence towards
the exact results can be obtained by using a small time increment.
In the calculations the atomic system of units is used. The units and conversions are:
,
where
,
e is the elementary charge and m the mass of electron
Mass unit:
|
1
au[mass] =
|
m
=
|
9.10953
10-31 kg
|
| Length
unit:
|
1
Bohr =
|
hBar^2
/ ( m e^2 ) =
|
5.28446
10-11 m
|
| Time
unit:
|
1
au[time] =
|
hBar^3
/ ( m e^4 ) =
|
2.41220
10-17 s
|
| Energy
unit:
|
1
Hartree =
|
m
e^4 / hBar^2 =
|
4.37189
10-18 J
|
1 Bohr = 0.0528446 nm
1 Hartree = 27.28700 eV
The results of calculation for a one dimensional wave packet
corresponding to a free particle are displayed at different time
instants. The time instants were chosen as integer multiples (in the range from
0 to 5) of time required to double the initial width of the wave packet.
The wave packet located at t=0 at x=0 has zero average momentum
and an initial width of
.
On the diagrams the unit of x is given in Bohr radius; the unit of the
wave number k is given in Bohr-1.
Fig. a.
Evolution of the real part of the wave function
.
Fig. b.
Evolution of the probability density of
the wave packet
.
The changing in time of the phase of the wave function causes the
oscillations in the real part of the wave function.
As k0=0,
the wave packet does not propagate, it is spreading out only, its width
Delta x is increasing with time t as given by formula (15).
Fig. c.
Real part of the Fourier transform of the wave function
.
Fig. d.
Square of the modulus of the Fourier transform of the
wave function
versus the wave number k at the same instants as in Fig.
a. and Fig. b.
At t=0 the functions are centred at k0=0 and have a
width of
.
Fig. d. clearly demonstrates the fact that the shape of the function
(the square of the modulus of the Fourier transform of
)
of a free particle does not change in time. As Delta x is spreading out
in the coordinate space, the wave function's modulus does not change in the
Fourier transform space (k space), the dispersion Delta k remains
the same; the function is not shrinking, showing that the Heisenberg
uncertainty relation is an inequality. The oscillations in the real part of
g(k,t) are due to the factor of
in equation (16).