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Indian Institute of Technology, Guwahati: T I T R H

This document discusses the Schrodinger wave equation and its application to solving problems of a particle in a one-dimensional box. It provides the time-dependent and time-independent Schrodinger equations. For a particle confined in a one-dimensional box between x=0 and x=L, the solutions for the wavefunctions and energy levels are derived. The energy is quantized and takes the form En = (n2h2)/(8mL2) where n is an integer representing the energy level. Figures 1 and 2 illustrate the energy levels/wavefunctions and probability densities of finding the electron at different positions in the box.

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Mihir Kumar Mech
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63 views4 pages

Indian Institute of Technology, Guwahati: T I T R H

This document discusses the Schrodinger wave equation and its application to solving problems of a particle in a one-dimensional box. It provides the time-dependent and time-independent Schrodinger equations. For a particle confined in a one-dimensional box between x=0 and x=L, the solutions for the wavefunctions and energy levels are derived. The energy is quantized and takes the form En = (n2h2)/(8mL2) where n is an integer representing the energy level. Figures 1 and 2 illustrate the energy levels/wavefunctions and probability densities of finding the electron at different positions in the box.

Uploaded by

Mihir Kumar Mech
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Indian Institute of Technology, Guwahati

Guwahati, INDIA 781 039


Department of Chemistry

Date: 01 August 2011


CH101
Class 3; Physical Chemistry

Schrdinger Wave Equation


Time dependent Schrdinger Wave Equation

H (r , t ) = i
t
Time independent Schrdinger Wave Equation

=E

is a function of spatial coordinates (x, y, z) or (r, , ) for a particular energy state (E).

H is known as the Hamiltonian operator or simply Hamiltonian.


H = (Kinetic Energy + potential Energy) operators = T + V

Typically, T =

2
2
2
2
p2
=
( 2 + 2 + 2 ) in three dimensions
2m x
2m
y
z

2 d2
p2
x

=
In one dimension, T =
2m dx 2
2m
V depends on the system and interaction involved.
Ze 2

For Coulombic potential: V =


(between an electron and nucleus with charge Z)
4 0 r
For Harmonic Oscillator potential: V = kX2

Solving Problems: Particle in a BOX. One-Dimensional Box. The particle is an


electron and it does not have enough energy to climb the barrier; therefore remains
confined to the box. What are the energy states that the particle can occupy?

2 d2
2m dx 2

Hamiltonian, H =

V =

V =

=E
2

Or,

d 2

=E
2m dx 2

Or,

d 2
2mE
= 2 = k 2 ;
2
dx

where k =

V=0

(2mE )1 / 2

Solutions:

X=0

X=L

= Aeikx ; = Be ikx and = Aeikx + Be ikx

However, = Ae

ikx

+ Be ikx is the most general solution.

= Aeikx + Be ikx , however, can be rewritten as = C cos kx + D sin kx


Boundary conditions:

= 0 and at x = L;

At x = 0;

When x = 0 ;

= 0.

= 0; then = C cos kx + D sin kx = C = 0

Hence, = D sin kx
However, at x = L;

= 0; then = D sin kL = 0

This is possible only when kL = n; where n = 1, 2, 3, 4


Hence, = D sin

nx
; the solution of the equation under the above boundary
L

conditions. One can also write ( x) = D sin

nx
L

However, D is still unknown. How to find D?


Since the particle must be inside the box, the total probability of finding the particle
inside the box must be unity.
2

* ( x) (x) dx = 1

Thus
0
L

D sin
0

2
L

D=
( x) =

Hence,

nx
nx
D sin
dx = 1
L
L

2
nx
sin
; complete solution.
L
L

One can see the solution is expressed in terms of the dimension of the box (L), position
(x) and quantum numbers (n=1, 2, 3, ).
One can derive the expression for energy of the particle in the box from
2

d 2

=E
2m dx 2
With the wavefunction ( x ) =

2
nx
sin
;
L
L

The expression for energy is

n2h2
En =
; where n is the quantum numbers, n = 1, 2, 3, 4,..
8mL2
Home work:
Find the average position of the particle in the one-dimensional box with the length L.
Hint: Evaluate the following integral
L

x = ( x) * x ( x)dx
0

Figure 1.

Energy and Wavefunctions of particle in a one-dimensional box (first

four states shown).

Figure 2. The probability densities of the electron inside the box at various location
in x and for different energy states.

n=5

(x)

n=4

n=3
n=2
n=1
X=0

X=L
4

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