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From the previous discussion on the double slit experiment on electron we found
that unlike a particle in classical mechanics we cannot describe the trajectory of an
electron.
We can however associate a wavefunction with each electron which will tell us the
probability amplitude of finding out an electron at a given point in space at a given
point in time.This wave function or probability amplitude has to be complex to
account for the double slit interference pattern created on the screen since the
double slit pattern is not a simple scalar some of the pattern coming out of the
individual slit.We shall denote this probability amplitude or wave function as Ψ(x,t)
The exact functional form of this wave function will depend on the potential under
which the electron is moving.Also it is now clear that since we cannot determine the
trajectory of a single electron in Quantum mechanics we shall not be interested in
finding out the time derivative of position or time derivative of velocity .
But we shall be interested in finding out the time derivative of the wavefunction it
self namely dΨ/dt. Thus we shall try to figure out which law calculate this quantity.
dt
dx
dt
dp
mdt
dv xx 1
Position and Momentum• From Classical Mechanics we find that to know all physical quantities we
need to find out the position and momentum.
• In Quantum mechanics correspondingly we need to find out the probability
distribution of position and momentum.
• In the previous section we found out that the wavefunction gives us the
probability of finding a quantum mechanical particle at a given point of
space at a given point time. What about the momentum?
• Let us take a special candidate for such wave function namely plane wave
something you have already studied in E.M. theory. In one spatial
dimension such a wave function is written as Ψ(x,t)=eikx where k is the
wavenumber. The probability density gives the intensity of the electron
wave on the screen is |Ψ(x,t)|2=1.This means that the probability is
uniform and the position is completely uncertain. Remember we are
talking of the same electron and after coming through a given slit under
any condition they can end up anywhere on the screen.
Momentum of a plane wave
• The wave number associated with such wave function is k. According to
the de Broglie hypothesis then the momentum associated with electron
is simply given by p= ħk . This is a remarkable result. It just shows
that while the postion of electron given by such wave function is
completely uncertain it has a definite momentum .
• Moreover a simple mathematical trick tells us that that above result can
be written as in the following way -iħ∂Ψ(x,t)/∂x = ħkΨ(x,t)=pΨ(x,t)
• The above result tells us something very important. At least for plane
wave like wave function if we operate it by a differential operation which
is in this case the first spatial derivative we get the same wave function
multiplied by the value of its momentum. Thus there exist some
connection between the x-component of the momentum p and the
differential operator -iħ∂/∂x
Momentum (Contd.)• Using a more complete and rigorous mathematical theory which we cannot
unfortunately describe in this course ( but generally taught in a full course
on Quantum Mechanics) actually it can be shown that indeed -iħ∂/∂x
represents the momentum operator along x-direction.
• The above statements means, for example if we want to know the
momentum associated with the wavefunction ( probability amplitude) of an
electron we need to operate the function with the above operator.
• This brings us to a strange fact again which we need to interpret. Suppose
the wavefunction of the electron is not given by a single plane wave but a
combination of many such plane wave. There is theorem in mathematics
which you may know. This is called Fourier theorem. It actually tells that
any well behaved function can be created by adding up a large no. of plane
waves with different values of wave number ( or wave vector) k .
• We shall particularly consider two such cases Ψ(x,t)=eik1x+ik
2x and
Ψ(x,t)=Aexp(-x2/σ²).
Momentum eigenfunction
• In both cases if we operate the wave functions with the operator -iħ∂/∂x we
do not get the same wave function multiplied by a definite value of the
momentum back , namely -iħ∂Ψ(x,t)/∂x ≠ pΨ(x,t). What will be the
momentum of the electrons with which such wavefunctions can be associated?
• The answer to this question is unlike the electron with which we associate a
single plane wave like wavefunction , these electrons do not have a defnite
wave number k or definite momentum p . They exist actually in a mixture of
electronic wavefunctions each of which has a specific momentum (plane
wave). In the first example the wave function composes of two such
momentum values since it is a combination of two plane waves. In the second
case using the mathematical theorem mentioned earlier it can be shown that
the wave function is a combination of infinite number of plane waves.
• When we try to measure the momentum of such electrons we ended up getting
these different values of composing momentums at different time and we call
their momentum uncertain.
• The case of the Gaussian wave function is even more interesting. Here the
function is localized within the length . therefore the uncertainty in finding the
electron position gets reduced as compared to plane wave. However this state as
we have mentioned can be mathematically written summing up a large number of
plane waves. Thus its momentum gets very uncertain. A comparison of this
wavefunction with the plane wave like wavefunction indicates that if we are able
to increase the uncertainty in position by choosing a new wavefunction this
results in an increase of uncertainty in momentum
• We can now summarize the preceding discussion to get some important
conclusions.
• We can associate the following operators with the three component of
momentum namely px -iħ∂/∂x , py -iħ∂/∂y , pz -iħ∂/∂z.
. The wavefunction associated with a quantum mechanical state can be either in a
state of definite momentum ( plane wave) or in a state with mixture of various
values of momentum giving uncertainty in momentum.
• If the wavefunction has a definite momentum the action of the
momentum operator will retain the same wavefunction multiplied by
the value of that definite momentum. Such a wave function is called
the eigenfunction of the corresponding operator ( in this case the
momentum operator) and the value of the momentum that multiplies
the wavefunction is the momentum eigenvalue.
• If the wavefunction is not in a definite momentum state, but in a
superposition of large number of momentum states, then the action of
the momentum operator on such state will yield a different function
altogether. The measurement of momentum of this wavefunction will
yield different values at different time according to the composition of
wavefunction, but each time one will get a specific value of the
momentum.
• The analysis of the wavefunction also indicates that if for a given
wavefunction the uncertainty in the position becomes less, the
uncertainty in momentum correspondingly increases.
• The positional operators are just given by the co-ordinate variables
themselves, namely x,y,z
• In classical mechanics most of the physical quantities ( better known as
dynamical variables) such as angular momentum, kinetic energy,
potential energy can be written as function of position and momentum
variables.
• In quantum mechanics we can associate an operator with each such
dynamical variable by replacing the position and momentum variables
with their respective operator. This means
• The significance of replacing a dynamical variable by a operator is the
following. Now if a wavefunction is given operating the wavefunction by
the operator corresponding to a given dynamical variable ( such as
energy momentum) we can immediately find out if the wavefunction is
an eigenfunction of that operator or not.
),(ˆ),(x
ixOpxO x
• If the wavefunction is an eigenfunction of that operator then the
corresponding particle is going to have a definite value of the related
physical quantity. For example, if the wave function of an electron is an
eigenfunction of the z-component of the angular momentum operator then
it has a definite Lz. Same is true for energy, momentum etc.
• On the otherhand if the corresponding wavefunction is not an
eigenfunction then everytime one measures the corresponding physical
quantities on the same quantum mechanical particle under identical
condition one will end up with different numbers. The corresponding
physical quantity becomes uncertain.
• However we can learn the following things about such uncertain quantities.
The wavefunction gives the probability amplitude at a given space and
time.The corresponding probability density is given by the modulus square
of the wavefunction . Using this probability density we can calculate the
mean value and higher moments of any physical quantity associated with
the particle in the following way.
dtrOtr
dtr
dtrOtrO ),(ˆ),(*
),(
),(ˆ),(*2
• The probability density of finding an electron in a small volume element
d around the point r is given by
• Since the electron or any other quantum mechanical particle must be
found somewhere in the space the total probability can be normalized to
1.
• The expectation value of any dynamical variable is therefore given by
• Such an expectation value can be obtained by repeating the same
experiment for a large no. of times and taking the average. The
dynamical variable thus behaves like a random variable.
dtrtr ),(),(
Variance
•We know that for any such distribution, we can define other moments,
say
•We define the variance of O as
And the standard deviation as
This is the spread in the results observed when we make a large number
of independent measurements and can be used to quantify the
uncertainty in that particular physical quantity.
For example if we use the plane wave state as the wave function and
calculate the standard deviation in the momentum, we shall find it is 0.
Thus the momentum of that state is well defined
dtrOtr
dtr
dtrOtrO ),(),(*
),(
),(),(*2
2
2
2
222 )()( OOO
22 )( OOO
Uncertainty Principle
• We have already pointed out by taking a plane wave like wavefunction
and a localized Gaussian type of wavefunction that in the case of the
former the position is completely uncertain whereas the momentum is
definite and in the later case position has much more uncertainty , but
the momentum is no more definite.
• Explicitly calculating the standard deviation using the preceding
formulas the above statement can be easily verified.
• The message is that uncertainty in momentum and position along a
given direction are somewhat inversely related.
• Around 1925 Heisenberg expresses this fact mathematically by stating
we cannot measure a pair of variables like position and the associated
momentum with arbitrary accuracy in the same experiment
2
Xpx
Complementarity
•Thus Heisenberg’s principle says that quantum mechanics
imposes certain limits on the accuracy with which we can
observe the world.
•A pair of variables like position and its associated momentum
which we cannot observe accurately together are said to be
complementary variables.
•Other such complementary pairs are rotational angle and the
associated angular momentum.
•The product of each pair has dimensions =[h]!
Commuting Operators
If two quantities can be measured simultaneously without
disturbing each other, they are said to commute. For instance
we can measure the x component of position of an object
without disturbing its y component and then measure the y
component or vice versa. So we expect
Here I have used a crescent symbol to indicate the fact that the
x is a measurement of the x coordinate. So if the operators
commute we can measure them in any order.
yxxy
Non-Commuting Operators
If two quantities cannot be measured simultaneously without
disturbing each other, they do not commute. For instance we
can measure the x component of position of an object but that
disturbs the corresponding momentum and vice versa. So we
expect
To achieve this in a way compatible with Heisenberg’s
uncertainty principle, we set
Thus
So we can write
The above quantity is called the commutator of x and px
xx pxxp xp
xipx
ixx
ix
ix
)(
ipxxpxp xxx ],[