Transcript chem6V19_postulates

Postulates of Quantum Mechanics
(from “quantum mechanics” by Claude Cohen-Tannoudji)
6th postulate: The time evolution of the state vector
is governed by the Schroedinger equation
 (t)
d
ih (t)  H(t)(t)
dt
where H(t) is the observable associated with the total energy of the system.


1st postulate: At a fixed time t0, the state of a
physical system is defined by specifying a ket

 (t 0 )
Postulates of Quantum Mechanics
(from “quantum mechanics” by Claude Cohen-Tannoudji)
2nd postulate: Every measurable physical quantity
is described by an operator
Qˆ .
Q
This operator is an observable.
3rd postulate: The only possible result of the
measurement of a physical quantity
 Q is one of the eigenvalues
of the corresponding
observable

Qˆ .
4th postulate (non-degenerate): When the physical quantity Q

is measured on a system in the normalized state  the probability of

ˆ is
obtaining the eigenvalue an of the corresponding observable Q
P an   un 
2
where
un
is the normalizedeigenvector of
associated with
 the eigenvalue


an .
Qˆ
Physical interpretation of
  
2
*

is a probability density. The probability of
finding the particle in the volume element
2

 x, y,z,t  dxdydz.

General solution for
 x, y,z,t 
Try separation of variables:
dxdydz
at time
n t   e
iEn t / h

and
is

 x, y,z,t    n x, y,z n t 
Plug into TDSE to arrive at the pair of linked equations:

t
Hˆ n  E nn
Orthogonality:
For
a , b
which are different eigenvectors of
we have orthogonality:
*

 ab  0
 bra/ket
Let us prove this to introduce the
notation used in the textbook

Hn  E nn