Dr.Eman Zakaria Hegazy Quantum Mechanics and Statistical Thermodynamics Lecture 10
Max Born
Max Born: a German physicist who was working actively in
scattering theory, found that this interpretation led to logical
difficulties and replaced Schrödinger’s with   (x ) (x )dx being the
probability that the particle is located between x and x+dx.
- From this we can use wave functions to calculate average
values and variances.

a  n



2

- The variance of x is x 2 n  3


2
2
1
2
n


- The variance of p is p
a
(17)
(18)
- But the uncertainty principle says that  p  x
(19)
2
- From Eqs. 17 and 18
 n

 2
2 3

 p x  
2
2
1
2
(20)
The value of the square- root term here is greater than unity
for all n
- We have used the simple case of a particle in a box to
illustrate some of the general principles and results of
quantum mechanics.
[1]
Dr.Eman Zakaria Hegazy Quantum Mechanics and Statistical Thermodynamics Lecture 10
The Postulates and General Principles of Quantum Mechanics
The state of a system is completely specified by its wave function
- Classical mechanics deals with quantities called dynamic
variables such as position, momentum, angular momentum,
and energy. A measurable dynamical variable is called an
observable.
- The classical- mechanical state of a one-body system at any
particular time is specified completely by the three position
coordinates (x, y, z) and the three momenta or velocities (vx,
vy, vz) at that time.
- The time evaluation of the system is governed by Newton’s
equations,
d 2x
m 2  Fx ,
dt
d 2y
m 2  Fy ,
dt
d 2z
m 2  Fz
dt
(21)
where Fx , Fy , and Fz are the components of the force,
F(x, y, z). Realize that generally each force component
depends on x, y, z. To emphasize this ,we write
d 2x
m 2  Fx (x , y , z ) ,
dt
d 2y
m 2  Fy (x , y , z ) ,
dt
d 2z
m 2  Fz (x , y , z )
dt
(22)
- Note that each of these equations is a second – order
equations, and so there will be two integration constants from
each one.
- We can choose the integration constants to be the initial
positions and velocities and write them as x0, y0, z0, vx0,vy0
,and vz0. The solutions to Eq. 22 are x(t), y(t), and z(t), which
describe the position of the particle as a function of time.
[2]
Dr.Eman Zakaria Hegazy Quantum Mechanics and Statistical Thermodynamics Lecture 10
- The position of the particle depends not only on the time but
also on the initial conditions. We write the solution of
equation 22 as:
x (t)=x(t; x0, y0, z0, vx0,vy0 , vz0)
y (t)=y(t; x0, y0, z0, vx0,vy0 , vz0)
z (t)=z(t; x0, y0, z0, vx0,vy0 , vz0)
- We can write these three equations in vector notation:
r (t)=r(t; r0,v0)
The vector r (t) describes the position of the particle as a
function of time; r (t) is called the trajectory of the particle.
- Classical mechanics provides a method for calculating the
trajectory of a particle in terms of the force acting upon the
particle through Newton’s equations 22
- The uncertainty principle is of no particle importance for
macroscopic bodies and so the classical mechanics is a
perfectly adequate prescription for macroscopic bodies ; but
for small bodies such as electrons , atoms , and molecules the
consequence of uncertainty principle are far from negligible
and so the classical mechanical picture is not valid.
- This lead us to our first postulate of quantum mechanics :
[3]
Dr.Eman Zakaria Hegazy Quantum Mechanics and Statistical Thermodynamics Lecture 10
Postulate 1:
The state of a quantum-mechanical system is completely
specified by a function ψ(r, t) that depends on the coordinate
of the particle and on the time. This function called the wave
function or the state function has the important property
that ψ*(r, t) ψ(r, t) dx dy dz is the probability that the
particle lies on the volume element dx dy dz , located at r , at
the time t.
- Postulate 1 says that the state of a quantum –mechanical
system such as two electrons is completely specified by
this function and nothing else is required.
- Because the square of the wave function have a
probabilistic interpretation a wave function must be
normalized so that in the case of one particle we have for
all time.





 (r , t ) (r , t )dxdydz  1
(23)
- For the specific case of a particle in a box, the limits are
(0, a). we can abbreviate equation 23 by letting dx dy dz=dτ
and to write



 (r , t ) (r , t )d   1

[4]
(24)