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Lecture 4. : The Free Particle
The material in this lecture covers the following in Atkins.
11.5 The informtion of a wavefunction
(a) The probability density
Lecture on-line
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Tutorials on-line
The postulates of quantum mechanics (This is the writeup for Dry-lab-II)( This lecture
does not cover any specific postulate)
Standing Wave (animation) (a must)
The wave Packet as superposition of plane waves (annimation) (a must)
A complete walk-through the free paricle (a must)
The Development of Classical Mechanics
Experimental Background for Quantum mecahnics
Early Development of Quantum mechanics
Audio-visuals on-line
The dual nature of matter (Quick Time movie 9 MB from Wilson group, *** )
Linear polarized light ( a wave function in 1-D would propagate in a similar way)
(1 MB Quick time movie from the Wilson Group, *****)
Circular polarized light ( ( a wave function could propagate in a similar way)
(6 MB Quick time movie from the Wilson Group, *****)
Slides from the text book (From the CD included in Atkins ,**)
The Classical Hamiltonian
Review
Consider a particle of mass m that is moving
in one dimension. Let its position be given by x
X
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O
In classical mechanics the state of a particle moving in
1 - D with the potential energy V(x,t) is determined completely
from the initial conditions at t= t o :
dV
d 2x
x(t o ) ; v(t o ) and  m 2 (Newtons Law)
dx
dt
The Classical Hamiltonian
Review
The expression for the total energy in terms of the potential
energy and the kinetic energy given in terms of the linear
momentum:
E  Ekin  Epot
p2

 V(r )
2m
is called the Hamiltonian:
p2
H
 V(r )
2m
Quantum Mechanics
Review
X
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The state of the same system in quantum mechanics is
determined by the time dependent Schrödinger equation:
(x,t) ˆ

 H(x,t)
i t
according to postulate 6
(x, t)
 (x, t)


 V(x, t)(x, t)
2
i
t
2m x
2
2
The wavefunction (X,t) contains
all the information about the
system according to postulate 1
Stationary States
Review
The total wavefunction for a one-dimentional particle in
a potential V(x) is given by
E
(x, t)  f(t)(x)  AExp[i t](x)
Where  (x) is determined from the time
independent Schrödinger equation
ˆ (x)  E(x)
H
Or :
2  2 (x)

 (x)V(x)  E(x)
2
2m x
Energy of system
Quantum Mechanical Principles..the Free Particle
The equation for (x) is given by
 2 (x)

2   (x)V(x, t)  E (x)
2m x
Let us now assume that V(x) = 0
2
In that case :
 (x)

2  E (x)
2m x
2
2
Quantum Mechanical Principles..the Free Particle
 (x)

2  E (x)
2m x
2
2
with the general solution :
 (x)  Aexp
ikx
 Bexp
ikx
since :
 2 [ Aexpikx  B expikx ] k 2 2
ikx
ikx


[Aexp

Bexp
]
2
2m
2m
x
2
 (x)
or : 
2  E (x)
2m x
2
2
2 2
k
E
2m
Quantum Mechanical Principles..the Free Particle
2  2 (x)
2 2
k


E

(x)
E
2
2m x
2m
 (x)  Aexp
ikx
 Bexp
ikx
Since the particle only has kinetic energy we must have
E
2
2
k
2m

2
p
2m
or : p = k
Quantum Mechanical Principles..the Free Particle

i 

ikx
ikx
(x, t)  Exp  Et Aexp  Bexp
)


Time face factor f (t).
Note f (t) f (t)  1
*
Has both real part
Acos kx  Bcos kx
and imaginary part
i( Asinkx  Bsinkx)
Quantum Mechanical Principles..the Free Particle
We get for the probability density:
i 
i 


(x, t)(x, t)  Exp  Et Exp Et 




ikx
 ikx
ikx
ikx 1
Aexp  B exp
) Aexp
 Bexp )
*


Or
(x, t)(x, t)  A exp
*
2
ikx
AB(exp
i2 kx
 B exp
exp
2
ikx
 exp
ikx

i 2kx
)
exp
ikx
Quantum Mechanical Principles..the Free Particle
(x, t)(x, t)  A exp
*
2
ikx
AB(exp
i2 kx
 B exp
2
1
exp
ikx
 exp
ikx
1

i 2kx
exp
ikx
)
| (x, t) |  A  B  AB[cos2kx  i sin 2kx] 
AB[cos2kx  i sin 2kx] 
2
2
2
(x, t) (x, t)  A  B  2 ABcos 2kx
*
2
2
Quantum Mechanical Principles..the Free Particle
(x, t) (x, t)  A  B  2 ABcos 2kx
*
2
2
B0
 i 
 (x, t)  AExp  Et Expikx


 i 
 (x, t)  AExp  Et 


(cos kx  i sin kx)
2
*
 (x, t) (x, t)  A
Same probability everywhere
Quantum Mechanical Principles..the Free Particle
i
Et
 (x, t)  AExp  Et Expikx   AExp i(kx  )




 (x, t)  Acos(kx 
Et
)  iAsin(kx 
Et
)
Re  (x, t)  Acos(kx 
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Et
)
Quantum Mechanical Principles..the Free Particle
Let us assume that at :
E
 Thus Acos(kx - Et )  0
t = to we have : kx - to =
2
At the later time :
E
E
E
 E
t = to  t kx - t = kx - t0  t   t
2
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E
However at : x +
t
k
we have at : to + t that
E
E
E

k(x + t)  (to  t)  kx - to =
k
2
E
The node has traveled x =
t
k
in t
Re  (x, t)  Acos(kx 
Et
)
Quantum Mechanical Principles..the Free Particle
E
The node has traveled x =
t
k
in t
E
velocity of the node is x/dt =
k
k2 2
E
p
mv v
x/dt =




k 2m k 2m 2m 2
2 2
k
E
2m
p = k
Re  (x, t)  Acos(kx 
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Et
)
Quantum Mechanical Principles..the Free Particle

i 

ikx
ikx
(x, t)  Exp  Et Aexp  Bexp
)


(x, t) (x, t)  A  B  2 ABcos 2kx
*
2
2
A 0
 i 
 (x, t)  BExp  Et Expikx 


 (x, t) * (x,t)  B2
 i 
 (x, t)  BExp  Et 


(cos(kx)  i sin(kx))
Quantum Mechanical Principles..the Free Particle
 i 

Et 
 (x, t)  AExp  Et Expikx   AExp i(kx  )




 (x, t)  Bcos(kx 
Et
)  iBsin(kx 
Et
)
Re  (x, t)  Bcos(kx 
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 Bcos(kx 
Et
)
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v
x/dt = 
2
Et
)
Quantum Mechanical Principles..the Free Particle
AB
(x, t) (x, t)  A  B  2 ABcos 2kx
*
2
2

i 

ikx
ikx
(x, t)  Exp  Et Aexp  Bexp
)


i 

(x, t)  AExp  Et (Expikx   Expikx )


    
Quantum Mechanical Principles..the Free Particle
    
i 

o (x,t)  AExp  Et (cos kx  i sin kx  cos kx  i sin kx)


i
o (x,t)  2AExp  Et cos kx


| o (x, t) |2  2 A2 (1  cos 2kx)
| o (x, t) |2  4A 2 cos2 kx)
Quantum Mechanical Principles..the Free Particle
i 

o (x,t)  AExp  Et (Expikx   Expikx )


Et 
Et 


o (x,t)  A(Exp i(kx 
 Exp i(kx 
)




Et
Et
o (x,t)  A(cos(kx  )  i sin(kx  )


Et
Et 

 A (cos(kx  )  i sin(kx  )


o (x,t)  2A cos(kx)cos(

Et
)  i cos(kx)sin(
Et 
)

Quantum Mechanical Principles..the Free Particle
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Re  (x, t) 
i 

Re AExp  Et Expikx


Et
 Acos(kx  )
Nodes move to the right
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Re  (x, t) 
i 

Re AExp  Et Expikx


Et
 Acos(kx  )
Nodes move to the left
Re  (x, t)  Acos(kx 
Et
)
Re  (x,t)  Acos (kx 

Et 
Re o (x, t)  2 Acos(kx)cos( ) Re   Re 


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Nodes do not move
Et
)
What you should learn from this lecture
1. For a free particle [ V(x) = 0] the time- independent
Schrödinger equation is given by :
2  2 (x)

 E(x)
2
2m x
with the general solution :
(x)  A expikx  Bexpikx
k2 2
and energy E given by E =
2m
You should also understand why physical arguments
require that the linear momentum must be p =  k
What you should learn from this lecture
2. For B = 0 ;   (x)  A expikx
This wave function represents a particle moving in the
positive x - direction with a constant probability density
  (x)*  (x) = A. We shal later show that
the particle has the momentum p= k
3. For A = 0 ;   (x)  Bexpikx
This wave function represents a particle moving in the
negative x - direction with a constant probability density
  (x)*  (x) = B. We shal later show that
the particle has the momentum p= -k
4. For A = B ; (x)  A[expikx  expikx ]
This wave function represents a particle with a
probability density (x) * (x) = 4A2 cos2kx. We shall
later discuss what state the particle is in
5. You should understand Figure 11.21 (page 300 in Atkins)