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Physics 451
Quantum mechanics I
Fall 2012
Oct 8, 2012
Karine Chesnel
Quantum mechanics
Announcements
Homework this week:
• HW # 12 due Thursday Oct 11 by 7pm
A8, A9, A11, A14, 3.1, 3.2
Quantum mechanics
A friendly message from the TA to the students:
I have noticed in recent homeworks that more students quit to do
entire problem(s). They are either short in time or overwhelmed
by the length of the problems. It is understandable that this is an
intense course, and the homework is time consuming. And as it is
approaching the middle of the semester, all kinds of things are
coming. But please be strong and do your best to learn. If you are
really out of time, do as much as you can.
Anyway, we don't want students to give up.
Quantum mechanics
Review- Matrices
Physical space
k
Generalization (N-space)
k’
• Linear transformation
Matrix
j’
j
i
i’
 
T  Tij
 
• Transpose
~
T  T ji
• Conjugate
T *  T ji
• Hermitian conjugate
• Unit matrix
Old basis
New basis
• Inverse matrix
• Unitary matrix
Expressing same transformation T
in different bases
 
*
Homework- algebra
Pb A8 manipulate matrices, commutator
transpose A , Hermitian conjugate
inverse matrix B 1  1 C
B
det B
Pb A9
a†b  a b
ab†  a b
Pb A11
Pb A14
 A, B
A†
scalar
matrix
matrix product
 ST  ,  ST 
†
1
transformation: rotation by angle q, rotation by angle 180º
reflection through a plane
matrix orthogonal T  T 1
Quantum mechanics
Need for
a formalism
Wave function
Operators
 
Hˆ   H ij 
Vector
Linear transformation
(matrix)
Quantum mechanics
Formalism
N-dimensional space: basis
Norm:
a 
e
1
, e2 , e3 ,... eN

a a
Operator acting on a wave vector:
Expectation value/ Inner product
T a  b
T  aT a
T  a T a  T †a a
b T a  T †b a
If T is Hermitian
b T a  Tb a
Quantum mechanics
Hilbert space
Infinite- dimensional space
N-dimensional space

e1 , e2 , e3 ,... eN



Wave function are normalized:

,  2 ,  3 ...  n ...
1
 ( x) dx  1
2

b
Hilbert space: functions f(x) such as

f ( x) dx  
2
a
Wave functions live in Hilbert space
Pb 3.1, 3.2
Quantum mechanics
Hilbert space

f g 
Inner product

f * ( x) g ( x) dx


Norm
f
2
 f f 

f * ( x) f ( x)dx

f m f n   nm
Orthonormality
f g  f
Schwarz inequality



g

f ( x)* g ( x)dx 



f ( x) dx  g ( x) dx
2
2

Quantum mechanics
Determinate states
Stationary states – determinate energy
H  n  En  n
Generalization of
Determinate state:
Standard deviation:
   Q  Q
2

2
For determinate state
For a given operator Q:
Q q
 2  Q2  Q
2

Q  Q
   Q  Q   Q  Q    Q  Q 
Q 
 0
Q

Q  Q 
operator

2
eigenvalue
eigenstate
0
2
Quantum mechanics
Hermitian operators
Observable - operator

Q 
Expectation value

 *Qdx   Q 

since
Q  Q
*
For any f and g functions
 Q    Q† 
f Qg  Qf g 
Q†  Q
Observables are Hermitian operators
Examples:
x̂
p̂