The motion of a quantum particle in an infinite one-dimensional potential well is discussed in these slides.

1. Particle in 1-Dimensional Infinite Potential Well
Schrödinger equation for a quantum particle
 Solutions of Schrodinger’s eq. in 1-D infinite potential well
 Energy eigen values and eigen function/wave functions
 Probability density distribution
 Quantum effect (How QP is different than CL particle)
Class objectives

2.  Einstein Quantum hypothesis: Light is not only a wave (EMW) but also a particle (discrete
energy packets, photon)
 de Broglie’s Hypothesis: Matter must also exhibit both particle and wave aspects
 Energy and momentum of particle are related to the matter wave parameters (average
frequency and wavelength)



E k
p 

 Observables can be deduced from the for a quantum particle
 Wavefunction contains all the information of quantum particle
)
,
( t
r

)
,
( t
r

Brief overview of the matter wave
 Particles is represented by
a group of waves (wavefunction)
 Use Schrodinger equation to get the wavefunction of a quantum particle
having motion in a defined potential region (not for all potentials)
Solve the Schrodinger equation for a free particle confined in a infinite potential well
Single wave associated to particle

3. L
m
0
m
0 L
V (x)
x
 m is mass of quantum particle
 Particle is confined in infinite potential well (1-D)
 Defined wave function for particle is )
(x

)
(
)
(
)
(
)
(
2 2
2
2
x
E
x
x
V
x
x
m









Time- independent Schrödinger equation
Solution of Schrodinger equation for quantum particle

4. )
(
)
(
2 2
2
2
x
E
x
x
m
n
n
n







For region (between boundaries)
0
)
( 
x
V
0
)
(
2
)
(
2
2
2




x
mE
x
x
n
n
n



0
)
(
)
( 2
2
2




x
k
x
x
n
n
n


2
2 2

n
n
mE
k 
For regions and
L
x  0

x


)
(x
V
0
)
( 
x

(1) where
Particle is free in the potential well, it changes direction at boundaries
We will have multiple solutions for so we introduce label n
L
x 

0
Probability of finding particle is zero, therefore
)
(x


5. General solutions of second order equation (1) are-
)
cos(
)
sin(
)
( x
k
B
x
k
A
x n
n
n 


Use wave function continuity conditions at boundaries
0

n

(i) At x = 0
0

B 
n
L
kn 
L
n
kn


(2)
Now solution
L
x
n
A
x
n

 sin
)
( 
Use normalized condition to determine A
m
0 L
V (x)
x
L
A
2

Final solution
L
x
n
L
x
n

 sin
2
)
( 
#Note: n = 0 is not
a solution (for n = 0,
)
0
)
( 
x
n

n = 1,2,3,4…
1
sin
|
|
0
2
2
2
0

 
 dx
L
x
n
A
dx
L
L
n


(ii) At x = L
B
B
A
n 

 0
cos
0
sin

We get
0
sin 
 L
k
A n
n

0

n

(3)

6. Energy of quantum particle
2
2
2
2
2mL
n
En


 where n = 1,2,3..
2
2
2
1
2mL
E



1
E
1
2 4E
E 
1
3 9 E
E 
1

n
2

n
3

n
 Energy levels are discrete in nature
 Energy of quantum particle is quantized inside the potential well
 Minimum energy of particle 0
2 2
2
2
1 

mL
E


1
2 4E
E 
1
3 9E
E 
1
4 16E
E 
L
n
kn


2
2 2

n
n
mE
k 
Use the relations
2
2
2
2
L
n
kn


2
2
2
1
1
2
)
1
2
(
)
1
2
(
mL
n
E
n
E
E
E
E n
n









 
Energy spacing between energy levels
 Energy spacing decreases with increasing the length of potential well

7. Eigen functions of quantum particle
L
x
n
L
x
n

 sin
2
)
( 
L
x
n
L
x
n

 2
2
sin
2
|
)
(
| 
Probability density distribution
 For each value of quantum number n, there is a specific wavefunction
 Wavefunction crosses zero n-1 times in this region (n-1 nodes)
 Probability distribution is not uniform
1

2

3

2
1 |
)
(
| x

2
2 |
)
(
| x

2
3 |
)
(
| x

n
L
2


 Wavelength of matter wave depends on L and n ( )

8. Classical Particle:
0
min 
E
 Particle can be stationary in the potential well,
2
2
1
mv
E 
0

v
 Energy of particle could be continues
Quantum Particle:
 Particle can not be stationary in the potential well,
0
2 2
2
2
min 

mL
E


 Energy of particle is discrete
1
4
1
3
1
2
16
9
4
E
E
E
E
E
E



 Probability of finding particle is constant at every position (x-value)
 Probability of finding particle is not uniform or constant (probability is zero at nodes)

9. Outcomes of class
Quantum particle can be represented by a wave function or wave packet in a finite region
 Energy of quantum particles is quantized
 Quantum particle will always be in motion between potential well boundaries
 Probability of finding particle is not constant but getting n-1 nodes
 Quantum particles characteristics are different than Classical particles
 For higher value of quantum number n, , particle will follow CL law’s of motion
0
2


n
L

Thank you for your kind
attention