Notes on quantum mechanics

1. 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
m
dt
dv x
x 1


2. 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.

3. 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

4. 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)=eik
1
x+ik
2
x and
Ψ(x,t)=Aexp(-x2/σ²).

5. 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.

6. • 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.

7. • 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.

8. • 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
i
x
O
p
x
O x



 

9. • 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.

10. 

 

 







d
t
r
O
t
r
d
t
r
d
t
r
O
t
r
O )
,
(
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.


 d
t
r
t
r )
,
(
)
,
(


11. 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


 

 







d
t
r
O
t
r
d
t
r
d
t
r
O
t
r
O )
,
(
)
,
(
*
)
,
(
)
,
(
)
,
(
* 2
2
2
2 




2
2
2
)
(
)
( O
O
O 






2
2
)
( 





 O
O
O

12. 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



 X
p
x

13. Heisenberg Microscope

14. 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]!

15. 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.

 y
x
x
y






16. 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

 x
x p
x
x
p





x
p
x
i
px



 


 

 i
x
x
i
x
i
x 


















 )
(

i
p
x
x
p
xp x
x
x 

 ]
,
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