9 problems (part-I and II) and in depth the ideas of Quantum; such as Schrodinger Equation, Philosophy of Quantum reality and Statistical interpretation, Probability Distribution, Basic Operators, Uncertainty Principle.

1. CONCEPTS & PROBLEMS IN QUANTUM MECHANICS-II
By Manmohan Dash

2. On Left; the electron of the Hydrogen atom
observed for the first time.
According to the wave function formalism
the wave function is an amplitude.
The square of this amplitude is “probability”
per unit of some quantity.
Probability; that the electron would be found
in a given range of that quantity, eg location
probability.
ψ

3. 4
Top

4. Wave Function ψ (x, t);
A function of (x) does not
give expectation value of
variables like p, in a direct
way.
Where; p: momentum.
There is a need to define
“operators” for such
variables.
Slide59, Part-I

5. For expectation value of
variables like (p) or their
function f(p);
That is; <p> and <f(p)>,
When wave function ψ = f(x, t).
Define operators for p.
Slide60, Part-I

6. Equations like Newton’s
Laws, of classical world,
“exist in quanta world”.
These equations relate
expectation values of
variables such as x, p
and V.
Slide61, Part-I

7. Heisenberg’s Uncertainty Relations;
expectation values of certain
variables not arbitrary with each
other;
e.g. < x > and <p> .
Their uncertainty ∆ and SD or σ
bear “inverse relationship”:
eg
2
~
x
x



p
p h

Slide62, Part-I

8. Representative problems

9. We saw that;
<x> is “average location” of the
Quantum, called as expectation
value of the “location of the
quantum”.
<x> is determined from wave
function or amplitude ψ(x, t).



 x|t)(x,|xx 2
d
How to know the expectation value
of momentum of the quantum? Or,
that of variable p.
In “classical world”, a particle has a
momentum p defined by its mass m
and velocity v;
p = m.v, that is, p is product of mass
and velocity.

10. So we can define velocity < v > ;
x
dt
d
v,x||xx 2
 


d
What is the velocity of the
quantum in a probabilistic
interpretation of wave
function?
In probability distributions; either discrete or
continuous, in part-I, we defined a central
tendency, called; mean or average or expectation
value of a variable such as x; < x >.
What is location x,
in probabilistic
interpretation of
wave function?

11. By using the “Schrodinger Equation (S)” and its complex
conjugate (S*) to evaluate;
 
 







)()()( **
tt
**
t
2
t
*
SS
2
t 

The probabilistic velocity leads to
the probabilistic momentum;
 x||xx 2
d
dt
d
m
dt
d
mvmp 
Lets take a step in that direction;
x)(x
2
i
xxmx xx
*
x
2
t
*
dd
dt
d
m  








  


12. Given
Step 1,
Step 2, Integration-By-Parts;
Wave Function properties ;
After two integration by parts;



 x|t)(x,|xx 2
d
x)(x
2m
i
xxx xx
*
x
2
t
*
dd
dt
d
 








  

|x
x
x
x
)(
x
)(
x
)(
x
b
a
b
a
b
a
fggd
d
df
d
d
dg
f
gf
d
d
g
d
d
fgf
d
d



 x,0
xx x
*
 

 d
m
i
dt
d
v 



13. Given < x >, after “integration by parts” two times; we have < p >;
Compare <x> and <p> ;
Thus <p> = , momentum; <p>, operator;
xx
x|t)(x,|xx
x
*
2








di
dt
d
mvmp
d














x
xi
xxx
*
*
dp
d







xi

xi 




p

14. A quantum found at C, upon
measurement !
 Where was it located right
before measurement?
 3 philosophies; Realist,
Orthodox, Agnostic
Orthodox view, or Copenhagen Interpretation; it was no where, location
indeterminate prior to measurement, act of measurement brought it.
Wave Function Collapsed at C.
 Most widely accepted, among Physicists, most respectable view.
 Experimentally confirmed, supported by Bell’s arguments.
The above figure is from DJ Griffiths

15. WAVE
FUNCTION
COLLAPSE
WAVE
FUNCTION
EVOLUTION
From http://www.mysearch.org.uk/
From DJ Griffiths

16. Agnostic view; we wouldn’t
know, “how many angels
on needle point”?
Rejected by Bell’s arguments.
Lack of experimental
support.
Realist or “hidden variable”
view; It was somewhere:
Present info not sufficient.
Deterministic. Not rejected
by Bell’s arguments, but
lack of experimental
support.
The mass of a quantum is spread out.
Wave Function of Copenhagen
Interpretation allows such distribution.

17. Does an
expecation value
give an average
value of a set of
measurement on
same particle?
EnsembleVs
One ParticleAverage is not in the sense of mathematics but
physics.
A value obtained on the first instance of measurement
is indeterminate, prior to measurement !
Wave function collapses after a measurement, any
further repetition of measurement, immediately,
would give same value as obtained before.
If its the same particle, it must go back to the state ψ
as it was prior to measurement, for any average to be
calculated, by further measurement.
Or an identically prepared ensemble of particles has
to be taken, all particles in the state ψ.

18. Problem 1.6
Why can’t you do
integration by parts
directly on middle
expression in
equation 1.29,
pull the time
derivative over onto
x, note that partial
of x wrt t is zero,
and conclude that
time derivative of
expectaion value is
zero.

19. Why can’t you do
integration by parts
directly on middle
expression in
equation 1.29,
pull the time
derivative over onto
x, note that partial
of x wrt t is zero,
and conclude that
time derivative of
expectaion value is
zero.
So we see that the integration does not
reduce to zero acording to the prescription
in the problem. 1.6 !

20. p is now an operator, so any general variable is simply
a replacement;
by the operator of p
Eg Q (x, p) has an expectation value given by;
Classically;
In probabilistic representation the operator of Kinetic
Energy T is;
In classical
mechanics all
variables can
be set as a
function of
location x and
momenta p.
In probabilistic
representation
also, these
variables can
be represented
through the x
and p variables.
xi 




p
 

 x)
xi
x,(p)x,( *
dQQ 

vmrLand
22
1 2
2

m
p
vmT
2
22
2
2
*
2
x2m
-
x
x2
-T
T 









d
m


21. Problem 1.7
Now let us discuss the Ehrenfest Theorem
as we have pointed out earlier. Let us
prove the theorem. Problem 1.7

22. Problem 1.7, from slide; 21

23. Problem 1.7, from slide; 21

24. Suppose you add a constant V0 to the potential energy (constant:
independent of x, t). In classical mechanics this does not chanage
anything. But what happens in Quantum Mechanics?
Show that the wave funcion picks up a time dependent phase factor
given below, what effect does this have on expectation value of
dynamic variables?
)t/iV-(x 0 pe
Problem 1.8

25. Lets see how to do
Problem 1.8

26. Remaining part of
Problem 1.8

27. Heisenberg Uncertainty relation There is a fundamental way
in which a classical wave
shows us that any
precision we have for
location of a point on the
wave comes from the fact
that waves can be
localized packets.
In that case the wavelength
and consequently the
momentum of the wave
become very badly
spread.
When we have a precise wavelength and
momentum (monochromatic wave) its quite
clear we wouldn’t know which location-point
of the wave would give a precise location of the
wave.
All the locus of the wave would suffice and we
would lose any sense of precise location.
Heisenberg Uncertainty relation

28. Heisenberg Uncertainty relation
This purely classical wave property
thus transpires to the
probabilistic systems that we have
been discussing so far.
We also note that momentum and
wavelength of a Quantum are
related as follows, this is called as
de-Broglie Relation; p = h/λ =
2πћ/λ
Thus in our probabilistic interpretations
the expectation values and
consequently the spread or error or
uncertainties given by standard
deviations of the distributions of
certain variables, bear an inverse
relation with each other.
This is called Heisenberg Uncertainty
Relation which we will discuss in more
rigor later. For now;
2
x

p

29. Lets discuss last problem of this presentation which
exemplifies some of the ideas we have discussed so far
including Heisenberg’s inequality of last slide.

30. Part (a)
Part (b)

31. Part (c)

32. Part (c)

33. Part (d)

34. So far we have discussed 9 problems in the last two
presentations and discussed in depth the ideas of
Quantum; such as Schrodinger Equation, Philosophy of
Quantum reality and Statistical interpretation,
Probability Distribution, Basic Operators, Uncertainty
Principle.
In the next lecture, we will discuss further problems that
will put us in a sound situation as regards a basic footing
in an introductory non relativistic quantum mechanics.
If you enjoyed the last two presentation styled lectures,
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students and teachers ,you know who could benefit from
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