Understanding Operator Formalism in Quantum Mechanics

7/17/2024
1
7/17/2024
2
Operator is a mathematical quantity when it operates
on one function, it charges the function into another one
and some times leave the function unaffected.
Examples of operators are addition, subtraction,
multiplication, division, differentiation, integration,
operations of grad, div, curl etc.
7/17/2024
3
 According to Heisenberg’s uncertainty principle,
some physical quantities like position, momentum,
energy, time, etc cannot be measured beyond a certain
degree of accuracy in quantum mechanics. Therefore,
the physical variables are given in terms of the  average
value. To determine the average of physical quantities,
some suitable operators are used.
7/17/2024
4
Among the so many measurements made on a
single dynamical variable, most of the time we can
get a particular value called the expectation value.
7/17/2024
5
 
 
 
 
The expectation value of an operator A is 
 
7/17/2024
6
Hamiltonian H = T + V
T = kinetic energy,   
V= potential energy 
 
 
      
 
We know that the value of momentum operator is 
 
From classical mechanics
7/17/2024
7
Free particle Hamiltonian
For a free particle V=0
 
7/17/2024
8
 
(
T
i
m
e
 
I
n
d
e
p
e
n
d
e
n
t
 
S
c
h
r
o
d
i
n
g
e
r
 
E
q
u
a
t
i
o
n
)
(
T
i
m
e
 
d
e
p
e
n
d
e
n
t
 
S
c
h
r
o
d
i
n
g
e
r
 
E
q
u
a
t
i
o
n
)
E= Energy Eigenvalue
 
 
 
 
7/17/2024
9
Prove that
 
The plane wave solution to the Schrodinger equation is
 
 
 
 
 
 
 
 
 
 
7/17/2024
10
 
 
or
 
 
 
7/17/2024
11
 
 
 
7/17/2024
12
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Eigenvalue equation is
 
 
 
 
 
 
 
 
 
7/17/2024 3:04:03 PM
13
 
 
 
 
 
 
 
 
7/17/2024
14
7/17/2024
15
 
 
 
 
 
 
 
 
 
 
 
 
7/17/2024
16
 
 
 
 
-1
  = 
 (or)  
-1
 = 
 
-1
 and 
-1
 
are inverse operators.
 
 
 
 
7/17/2024
17
The parity operator is a special mathematical operator and 
 
is denoted by    .
For a wave function of the variable x, The parity operator 
is defined as
 
This means that when the wavefunction 
by the parity operator, it gets reflected in its co-ordinates.
is operated 
7/17/2024
18
The Eigenvalue equation of the parity operator is
 
 
Operating the above equation again by 
,
 
 
 
 
This means that is the parity operator is operated twice 
 
Hence, 
 
7/17/2024
19
Therefore, the Eigenvalues are +1 and -1. 
F
rom the equations 1 and 2
 
 
 
If λ =1, the wave function is even
If λ = -1, the wave function is add
Bosans are described by symmetric wave function.
Fermions are described by symmetric wave function.
7/17/2024
20
 
 
 
 
 
 
ie
                   AB + BA =0
AB – BA = 0
  [A, B] = 0
Therefore
7/17/2024
21
An operator is said to be linear if it satisfies the relation 
 
where C
1
 and C
2
 are constants.
The inverse operator A
-1
 is defined by the relation 
An operator commutes with its inverse
 
7/17/2024
22
Consider the Pauli
s spin operator
 
 
Conjugate transpose is called dagger.
If 
Then
 
is Hermitian
7/17/2024
23
 
 
 
An operator 
 is said to be Hermitian if 
                                                                         =A and it should
satisfy the following condition.
7/17/2024
24
For any operator A
 
 
 
 
 
 
 
(a) Hermitian, (b) anti Hermitian, (c) unitary, (d) orthogonal
 
 
 
 
7/17/2024
25
Eigen values of Hermitian operators are real
Consider the Eigenvalue equation
 
Pre multiply equation 1 by 
𝚿
*
 and than integrate
 
 
If A is Hermitian 
 
 
 
 
From equations 2 and 3
 
 
 
 
 
 
7/17/2024
26
Two Eigenfunctions of Hermitian operators, belonging to different Eigenvalues,
are orthogonal
Consider 𝚿, and 𝜙 are the two Eigenfunctions of the Hermitian
operator   . There we can write
 
 
 
Pre multiply equation 1 by 
𝜙
*
 and then integrate, we can get
 
If A is Hermitian
 
 
 
 
 
 
7/17/2024
27
 
3-4 
 
 
From equation 5, it is clear that (a-b) ≠ 0
i.e a ≠ b, but 
                               should be equal to zero. 
This means
that the wavefunctions 𝜙 and 𝚿 are mutually 
orthogonal
.
 
 
7/17/2024
28
The product of two Hermitian operators is Hermitian if and only if they commute. 
Suppose 
𝚿
1
 and 
𝚿
2
 are two functions, using the operators A and B, we can
develop an integral
 
If A is Hermitian
Again, if 
 is Hermitian we can write
If AB is Hermitian
 
7/17/2024
29
If the operators A and B commute, we have
 
Which is the condition for the product operator 
to be Hermitian
7/17/2024
a) [A, B] = C,   b) AB + BA = C,   c) ABA = C,   d) A+B =C
Solution 
Given
 
 
 
 
 
+
 
 
 
 
 
 
 
 
 
 
 
 
 
7/17/2024
31
Prove that the momentum operator is Hermitian
 
 
If P is said to be Hermitian
 
The expectation value of 
 can be written as
 
 
We have to solve the above integral 
Put
 
   
    
Momentum operator 
7/17/2024
32
 
 
 
 
 
 
7/17/2024
33
Prove that parity operator commutes with the Hamiltonian
π = Parity operator
H = Total Hamiltonian 
We have to prove [π, H] = 0
ie       πH = H π
          πH - H π = 0
we know that 
 
 
7/17/2024
34
 
 
 
7/17/2024
35
 
 
 
 
 
7/17/2024
36
We know that the orbital angular momentum
 
 
7/17/2024
37
 
 
 
 
 
 
 
 
7/17/2024
38
 
 
7/17/2024
39
 
 
 
 
 
 
 
 
 
 
 
 
7/17/2024
40
 
 
 
 
 
 
7/17/2024
41
 
 
ll
iy
 
 
 
But 
 
 
 
7/17/2024
42
Value of 
 
 
Commutation relation between L
2
 and L
z
 
 
 
 
 
 
 
 
 
 
 
 
7/17/2024
43
We know that [AB,C] = [A,C]B + A[B,C]
 
 
In the same way we can prove that
 
 
 
 
    
     
 -----------1
 
7/17/2024
44
  -------------2
 
From the equations 1 and 2 
We may conclude that the square of the angular
momentum operator commutes with one of its components
but the components among themselves do not commute. 
7/17/2024
45
Raising and Lowering operators (Ladder operators)
                                                                                          
are called raising
and lowering operators, repectively.
Each time operation of the raising operator may increase the 
Eigenvalue of the system by one unit of ћ
.
  On the other hand, each time operation of lowering operator may 
   
decrease the 
E
igenvalue of the system by one unit of ћ     .
Therefore, these operators are called Ladder operators.
7/17/2024
46
Commutation relation between 
 
 
 
Similarly
 
7/17/2024
47
 
Commutation relation between 
 
 
 
 
 
 
 
 
7/17/2024
48
 
In the same way we can prove that 
 
7/17/2024
49
Slide Note
Embed
Share

Dive into the world of quantum mechanics with Dr. N. Shanmugam as he explains the role of operators, their significance in quantum mechanics, and how they are used to determine physical quantities through expectation values. Explore concepts such as the Hamiltonian operator, time-independent Schrodinger equation, and the plane wave solution. Gain insights into the fundamental principles that govern the behavior of particles at the quantum level.


Uploaded on Jul 17, 2024 | 2 Views


Download Presentation

Please find below an Image/Link to download the presentation.

The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. Download presentation by click this link. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.

E N D

Presentation Transcript


  1. OPERATOR FORMALISM OF QUANTUM MECHANICS Dr. N. Shanmugam ASSISTANT PROFESSOR DEPARTMENT OF PHYSICS ANNAMALAI UNIVERSITY DEPUTED TO D. G. Govt. Arts college (W) Mayiladuthurai-609001 7/17/2024 1

  2. WHAT IS AN OPERATOR? Operator is a mathematical quantity when it operates on one function, it charges the function into another one and some times leave the function unaffected. Examples of operators are addition, subtraction, multiplication, division, differentiation, integration, operations of grad, div, curl etc. 7/17/2024 2

  3. WHY WE NEED AN OPERATOR? According to Heisenberg s uncertainty principle, some physical quantities like position, momentum, energy, time, etc cannot be measured beyond a certain degree of accuracy in quantum mechanics. Therefore, the physical variables are given in terms of the average value. To determine the average of physical quantities, some suitable operators are used. 7/17/2024 3

  4. EXPECTATION VALUE Among the so many measurements made on a single dynamical variable, most of the time we can get a particular value called the expectation value. 7/17/2024 4

  5. The expectation value of an operator A is 7/17/2024 5

  6. Hamiltonianoperator From classical mechanics Hamiltonian H = T + V T = kinetic energy, V= potential energy We know that the value of momentum operator is 7/17/2024 6

  7. Free particle Hamiltonian For a free particle V=0 7/17/2024 7

  8. ( (Time Independent Schrodinger Equation) E= Energy Eigenvalue (Time dependent Schrodinger Equation) 7/17/2024 8

  9. Prove that The plane wave solution to the Schrodinger equation is 7/17/2024 9

  10. or 7/17/2024 10

  11. 7/17/2024 11

  12. 7/17/2024 12

  13. Eigenvalue equation is 13 7/17/2024 3:08:33 PM

  14. 7/17/2024 14

  15. Identity (or) Unity operator Null operator 7/17/2024 15

  16. Inverse operator (or) -1 = -1 = -1 and -1are inverse operators. Equal operator 7/17/2024 16

  17. Parity operator The parity operator is a special mathematical operator and is denoted by . For a wave function of the variable x, The parity operator is defined as This means that when the wavefunction by the parity operator, it gets reflected in its co-ordinates. is operated 7/17/2024 17

  18. , Eigenvalue of parity operator The Eigenvalue equation of the parity operator is Operating the above equation again by This means that is the parity operator is operated twice Hence, 7/17/2024 18

  19. Therefore, the Eigenvalues are +1 and -1. From the equations 1 and 2 If =1, the wave function is even If = -1, the wave function is add Bosans are described by symmetric wave function. Fermions are described by symmetric wave function. 7/17/2024 19

  20. Commuting operator Therefore AB BA = 0 [A, B] = 0 Anti commuting operator ie AB + BA =0 7/17/2024 20

  21. Linearoperator An operator is said to be linear if it satisfies the relation where C1 and C2 are constants. The inverse operator A-1 is defined by the relation An operator commutes with its inverse 7/17/2024 21

  22. Hermitianoperator Consider the Pauli s spin operator Conjugate transpose is called dagger. If Then is Hermitian 7/17/2024 22

  23. An operator satisfy the following condition. is said to be Hermitian if =A and it should 7/17/2024 23

  24. For any operator A (a) Hermitian, (b) anti Hermitian, (c) unitary, (d) orthogonal 7/17/2024 24

  25. Eigen values of Hermitian operators are real Consider the Eigenvalue equation Pre multiply equation 1 by ?* and than integrate If A is Hermitian From equations 2 and 3 7/17/2024 25

  26. Two Eigenfunctions of Hermitian operators, belonging to different Eigenvalues, are orthogonal Consider ?, and ? are the two Eigenfunctions of the Hermitian operator . There we can write Pre multiply equation 1 by ?* and then integrate, we can get If A is Hermitian 7/17/2024 26

  27. 3-4 From equation 5, it is clear that (a-b) 0 i.e a b, but should be equal to zero. This means that the wavefunctions ? and ? are mutually orthogonal. 7/17/2024 27

  28. The product of two Hermitian operators is Hermitian if and only if they commute. Suppose ?1 and ?2 are two functions, using the operators A and B, we can develop an integral If A is Hermitian Again, if is Hermitian we can write If AB is Hermitian 7/17/2024 28

  29. If the operators A and B commute, we have Which is the condition for the product operator to be Hermitian 7/17/2024 29

  30. If A, B, and C are non-zero Hermitian operators, which of the following relation must false? a) [A, B] = C, b) AB + BA = C, c) ABA = C, d) A+B =C Solution Given + 7/17/2024

  31. Prove that the momentum operator is Hermitian Momentum operator If P is said to be Hermitian The expectation value of can be written as We have to solve the above integral Put 7/17/2024 31

  32. 7/17/2024 32

  33. Prove that parity operator commutes with the Hamiltonian = Parity operator H = Total Hamiltonian We have to prove [ , H] = 0 ie H= H H - H = 0 we know that 7/17/2024 33

  34. 7/17/2024 34

  35. 7/17/2024 35

  36. Angular momentum operators We know that the orbital angular momentum 7/17/2024 36

  37. 7/17/2024 37

  38. 7/17/2024 38

  39. 7/17/2024 39

  40. 7/17/2024 40

  41. lliy But 7/17/2024 41

  42. Value of Commutation relation between L2 and Lz 7/17/2024 42

  43. We know that [AB,C] = [A,C]B + A[B,C] In the same way we can prove that -----------1 7/17/2024 43

  44. -------------2 From the equations 1 and 2 We may conclude that the square of the angular momentum operator commutes with one of its components but the components among themselves do not commute. 7/17/2024 44

  45. Raising and Lowering operators (Ladder operators) are called raising . and lowering operators, repectively. Each time operation of the raising operator may increase the Eigenvalue of the system by one unit of On the other hand, each time operation of lowering operator may decrease the Eigenvalue of the system by one unit of . Therefore, these operators are called Ladder operators. 7/17/2024 45

  46. Commutation relation between Similarly 7/17/2024 46

  47. Commutation relation between 7/17/2024 47

  48. In the same way we can prove that 7/17/2024 48

  49. 7/17/2024 49

Related


More Related Content

giItT1WQy@!-/#giItT1WQy@!-/#giItT1WQy@!-/#giItT1WQy@!-/#giItT1WQy@!-/#giItT1WQy@!-/#giItT1WQy@!-/#giItT1WQy@!-/#giItT1WQy@!-/#giItT1WQy@!-/#