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CONCEPTS & PROBLEMS IN QUANTUM MECHANICS-II
By Manmohan Dash
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
4
Top
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
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
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
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
Representative problems
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.
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?
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  








  

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 


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
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
WAVE
FUNCTION
COLLAPSE
WAVE
FUNCTION
EVOLUTION
From http://www.mysearch.org.uk/
From DJ Griffiths
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.
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.
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.
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 !
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

Problem 1.7
Now let us discuss the Ehrenfest Theorem
as we have pointed out earlier. Let us
prove the theorem. Problem 1.7
Problem 1.7, from slide; 21
Problem 1.7, from slide; 21
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
Lets see how to do
Problem 1.8
Remaining part of
Problem 1.8
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
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
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.
Part (a)
Part (b)
Part (c)
Part (c)
Part (d)
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,
leave me any note, feedback, errors to
g6pontiac@gmail.com , feel free to share this among
students and teachers ,you know who could benefit from
this. Also you can visit my website ; mdashf.org !!

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Concepts and Problems in Quantum Mechanics, Lecture-II By Manmohan Dash

  • 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
  • 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
  • 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
  • 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
  • 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.
  • 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, leave me any note, feedback, errors to g6pontiac@gmail.com , feel free to share this among students and teachers ,you know who could benefit from this. Also you can visit my website ; mdashf.org !!