Transcript EP-307 Introduction to Quantum Mechanics

The Propagator
The idea was not to solve the position of two masses coupled by three
Springs.
This is simply done as you have done in classical mechanics by
defining normal modes xII=(x1+ x2)/2 & xII=(x1- x2)/2
The idea was to pick up a simple problem and use all the mathematics
that you have learnt in the last one month to solve a physics problem
to write down the propagator. For once you write down the propagator
one has effectively solved the problem.
One had talked about
the basis,
basis transformations,
how one transforms the basis
using Unitary transformations,
elements of this Unitary Operator
eigenvalue and eigenkets
All of this were used in the problem
How does one move forward—
You know something and you apply it to something new
राघव वर्ाा
Last t imewe had writ t endown t hisequat ion
The Propagator
x (t )  U p x (0)
W here x(t) is t hefinalst at e vect orU p is t hepropagat or
and x (0) is t heinit ialst at e vect or.
In t ermsof t heeigenvalues and eigenket sof  t hesame
equat ion reads
x (t )   I I cos I t  II II cos II t  x(0)
T hequant it yin t heparent hesis we ident ifiedas
t he propagat orU p
Last t imewe wrot edown t heelement sof U p by t ransforming
t he vect orsin t heabove equat ion in t heoriginalbasis
W e can also writ e down t heelement sof U p direct ly
U p11  1 I I 1 cos ωI t  1 II II 1 cos ωII t
U p12  1 I I 2 cos ωI t  1 II II 2 cos ωII t
U p 21  2 I I 1 cos ωI t  2 II II 1 cos ωII t
U p 22  2 I I 2 cos ωI t  2 II II 2 cos ωII t
It is left as an exercisefor you t oshow t hatindeed t hisgives t he
same result as derivedin t heclass.
Furt her t he form of U in t heeigenbasis of  is
cos I t

Up  
cos II t 

One can now t ransformU direct ly o
t t heoriginalbasis.W e know
how t o do t hat
राघव वर्ाा
Up  U TU pU
The Normal Modes
T hereare t wo init ialst at es x( 0 ) for wh icht he t im eevo lut ion
is part icularly sim ple
Sup pose x( 0 )  I
T hen x(t)  I
 I
I I cos ωI t

II
II II cos ωII t
cos ωI t
T hus a syst emst art in goff in I is on ly m odifiedby an ov erall
fact orcos ωI t
Sim ilarly for a syst emst art in goff in x( 0 )  II t he t im e
evo lut ionis part icularly sim ple
x(t)  II
cos ωII t
In t hefirst case t he t wo m asses oscillat ein ph ase wit h t hespring
in t hem iddle hav in gno role

k 
cos
t

1
m 

x (t ) 
k 
2 
cos
t

m 


In t hesecon d case t hem iddle spring exert sa force2k & t he
spring nearest k. T he t ot aladds upt o 3k.
T heseare called Norm alMo des. In quan t um m echanicst hey
are also called t hest at io naryst at es.T hisis because a syst em
st art in gin on e of t hesest at esrem ainsin t hatst at efor all t im es.
T hatis it s st at edoes no t changeas a funct ionof t im eor is
st at io narywit h respect t ot im e. If t hesyst emst art sas a lin ear
com binat io
n it t henevo lv esas a lin ear com binat io
n. T hisst at e
changesas a funct ionof t im e.
T hecen t ralpro blemin quan t um m echanicsis verysim ilar.T he
st at eof t hesyst emis described in quan t um t heoryby t heket 
which obeysSch ro edinger's equat ion
i 
H
T hepro blemis t o find  (t ) given  (0) . Since t hisis a first
order different al
i equat ion ,no assum pt ionabo ut  (0)
is required.
राघव वर्ाा
Com in gback t oit 
quan t um syst em .
con t ain sall t heinform at io
n abo ut t he
Function of Operators & Related Concepts

Two types of quantities that operate on vectors
–
–
Scalars which commute with eact other and with operators
Operators which generally do not commute

First as C numbers and the second are known as Q numbers

Functions of c numbers such as sin x, ex, log(x) …

Functions of q numbers?
–
–
–
–

Would be f() = ann
The definition would make sense if it converges

Lets look at
n

e 
n 1
n!
Lets restrict ourselves to Hermitian operators, then in its eigenket basis we have
1 0
0 
2




0 0
राघव वर्ाा
0
 0 
 

 n 

 m1
0

0  m2
m  
 


0
 0
0 

 0 

 

  m n 

Function of Operators & Related Concepts
e1

0

e 
 

 0
0 

 0 
  

 en 

0
e 2

0
Since each sum converges to a familiar limit eI the operator e is then
   m1

m 0 m!

e   0
 

 0

0

 m2
m 0
m!


0




0 


 


 mn 
 
m  0 m! 


0
This is a finite quantity and hence we can
use this as an operator.
Derivatives of Operators w.r.t. parameters
Consider an operator  that depends on a parameter . Its derivative w.r.t

d( )
 (   )  ( ) 
 Lim 

 0
d


If  is written as a matrix in some basis, then the matrix representing
Is obtained by differentiating the matrix element 
राघव वर्ाा
d ( )
d
Function of Operators & Related Concepts
d( )
 e   e    ( )
d
Same result may be obtainedeven if  is not Hermitianby
considering thepower series if it exists


d   n  n   nn 1 n
n 1 n 1
m  m
 
  
 
 
d  n 0 n!  n 0
n!
m!
n 1 ( n  1)!
m 0
 e 
Converselyif we havea differential equat ion
d( )
 e 
d


( )  c exp  d    C exp 
0

where C is a constantof int egration
Here  behavesas C number.T hedifferencebetween c number
and q numbersis that while c numberscommuteq dont.However,
if only one q number is involvedtheneverythingcommuteswith
everything
.
T hingsare differentwhen more thanone q numbers,as then
order of factorsis important
e  e    e (   ) 
e  e    e
   
e e  e   e
राघव वर्ाा

unless ,    0