Particle in One-Dimensional Infinite potential well (box)

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 eigenvalues and eigenfunction/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 kp
Observables can be deduced from the for a quantum particle
Wavefunction contains all the information of quantum particle
),(tr
),(trBrief overview of the matter wave
Particles is represented by
a group of waves (wavefunction)
Use Schrodinger equation to get the wavefunctionof 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
22
xExxV
x
x
m







Time-independent Schrödinger equation
Solution of Schrodinger equation for quantum particle

4. )(
)(
2
2
22
xE
x
x
m
nn
n







For region (between boundaries)
0)(xV
0)(
2)(
22
2



x
mE
x
x
n
nn



0)(
)( 2
2
2



xk
x
x
nn
n


2
22

n
n
mE
kForregions and
Lx 0x
)(xV
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
Lx0
Probability of finding particle is zero, therefore
)(x

5. General solutions of second order equation (1) are-
)cos()sin()( xkBxkAx
nnn

Use wave function continuity conditions at boundaries
0
n
(i) At x= 0
0B nLk
n
L
n
k
n


(2)
Now solution
L
xn
Ax
n

 sin)(
Use normalized condition to determine A
m
0 L
V (x)
x
L
A
2

Final solution
L
xn
L
x
n

 sin
2
)(
#Note: n = 0 is not
a solution (for n = 0,
)
0)(x
n

n = 1,2,3,4…
1sin||
0
222
0

dx
L
xn
Adx
LL
n


(ii) At x= L
BBA
n
 0cos0sin
We get
0sin LkA
nn
0
n
(3)

6. Energy of quantum particle
2
222
2mL
n
E
n


where n = 1,2,3..
2
22
1
2mL
E


1E
124EE
13
9EE
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
22
1 
mL
E

12
4EE
13
9EE
1416EE
L
n
k
n


2
22

n
n
mE
k
Use the relations
2
22
2
L
n
k
n


2
22
1
1
2
)12()12(
mL
nEnE
EEE
nn




Energy spacing between energy levels
Energy spacing decreases with increasing the length of potential well

7. Eigen functions of quantum particle
L
xn
L
x
n

 sin
2
)(
L
xn
L
x
n


22
sin
2
|)(| 
Probability density distribution
For each value of quantum number n, there is a specific wavefunction
Wavefunctioncrosses 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
L2

Wavelength of matter wave depends on Land n ( )

8. Classical Particle:
0
minE
Particle can be stationary in the potential well,
2
2
1
mvE
0v
Energy of particle could be continues
Quantum Particle:
Particle can not be stationary in the potential well,
0
2
2
22
min

mL
E

Energy of particle is discrete
14
13
12
16
9
4
EE
EE
EE



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