Download - Schrodinger Equation
-
7/27/2019 Schrodinger Equation
1/21
-
7/27/2019 Schrodinger Equation
2/21
Schrodinger Equation
-
7/27/2019 Schrodinger Equation
3/21
Each particleis represented by awave function (x,t) such that *=theprobability of finding the particle at that
position at that time
The wave function is used in theSchrodinger equation. The Schrodinger
equation plays the role of Newtons lawsand conservation of energy in classicalmechanics. It predicts analytically andprecisely the probability of events.
THE WAVE FUNCTION
-
7/27/2019 Schrodinger Equation
4/21
Wavefunction Properties
contains all the measurable information aboutthe particle
* summed over all space=1(if particleexists, probability of finding it somewheremust be one and its derivative is continuous
allows energy calculations via the Schrodingerequation
permits calculation of expectation value of agiven variable.
is finite and single valued
-
7/27/2019 Schrodinger Equation
5/21
))(/( pxEtiAe
SCHRODINGERS EQUATION:
TIME-DEPENDENT FORM
The wave function for a particle moving freely in
the +x direction is given by.(1)
where E is total energy of the particle and p is its
momentum.Differentiating (1) for twice w r to x gives
Also differentiating (1) for once w r to tgives
2
2
22
2
p
xor
.(2)
iE
t .(3)or tiE
-
7/27/2019 Schrodinger Equation
6/21
),(2
2
txVm
pE
V
xmt
i2
22
2
.(4)
The quantities
are called operators.
The total energy E of aparticle is the sum of itskinetic energy and potentialenergy
Multiply (5) by on bothsides gives
Substitute for E and p2 to obtain the timedependent form of schrodinger equation:
.(6)
..(5)
.(7)
tiE
and
Vm
pE2
2
-
7/27/2019 Schrodinger Equation
7/21
V
zyxmti
2
2
2
2
2
22
2
In three dimensions the time dependent form ofSchrodingers equation is
.(8)
Where the particles potential energy V is some
function of x,y,z and t. Any restriction that maybe present on the particles motion will affect thepotential energy and once V is knownSchrodinger's equation may be solved to obtain the
energy, wave function and probability density ofthe particle.
-
7/27/2019 Schrodinger Equation
8/21
EXPECTATION VALUES
)1...(..................
.....
321
332211
i
ii
N
xN
NNN
xNxNxNX
)2.....(....................2dxP ii
The average position of a number of particlesdistributed along the x axis such that N1 particles
at x1, N2 particles at x2 and so on is given by
For a single particle the number Ni at xi must bereplaced by the probability
Where i is the particle wave function evaluatedat x=xi. Making substitution and changing thesummations to integrals, the expectation value ofthe position is given by
-
7/27/2019 Schrodinger Equation
9/21
)3..(..............................2
2
dx
dxx
x
If the wave function is normalized
then )4...(..............................2
dxxx
1..2
dxei
)5......(2
dxEE
dxpp2
Similarly expectation values of momentum andenergy are given by
and
-
7/27/2019 Schrodinger Equation
10/21
)1(.......................................... )/(
)/()/())(/(
tiE
xiptiEpxEti
eei
eAeAe
)2(..........2 2
22
V
xmti
tiEtiEtiE
eVxemeE
)/(
2
2)/(
2)/(
2
xipAewhere )/(
SCHRODINGER EQUATION: STEADY-STATE FORM
The wave function
of unrestricted particle may bewritten as
is the product of time dependent and a positiondependent function
Substituting in time dependent form ofSchrodinger equation
We get
-
7/27/2019 Schrodinger Equation
11/21
Dividing through by the common exponential factorgives
is the steady-state form of Schrodinger equation.
In three dimensions the equation takes the form as
)4........(0)(2
22
2
2
2
2
2
VE
m
zyx
)3(....................0)(
222
2
VE
m
x
-
7/27/2019 Schrodinger Equation
12/21
-
7/27/2019 Schrodinger Equation
13/21
EIGENVALUES AND EIGENFUNCTIONS
The values of energy En for which Schrodingers
steady-state equation can be solved are calledeigenvalues and corresponding wave functions nare called eigenfunctions.
The discrete energy levels of the hydrogen atom
are an example of a set of eigenvalues
2222
4 1
32 n
meEn
n = 1,2,3,..
-
7/27/2019 Schrodinger Equation
14/21
)1.(..............................02
22
2
E
m
x
APPLICATION: PARTICLE IN A BOX
0 L x
V
Aim: To show energy quantization
1.How to solve Schrodingers equation?
2.Obtain eigenvalues and eigenfunctions
3.Compare the results with classicalmechanics
The potential energy of the particle isV=0 inside the box andV= outside the box
The steady-state form of Schrodingersequation for the particle within the box with aboverestrictions may be written as
-
7/27/2019 Schrodinger Equation
15/21
)2...(2
sin xmE
A
)4......(..............................2 nLmE
The equation (1) will have the following possiblesolutions
)3......(2
cos xmE
B
and
These solutions are subject to the importantboundary condition such that =0 for x=0 and x=LSince cos(0)=1, the second solution cannot
describe the particle at x=0 implies B=0
The first solution gives=0 for x=0 but will bezero at x=L only when
n=1,2,3,..
From equation (4) the energy of the particle canhave only certain values called eigenvalues givenby
-
7/27/2019 Schrodinger Equation
16/21
)7...(..........)sin(2
sin xL
nAx
mEA
n
n
)5(..............................2 2
222
mL
nEn
n=1,2,3,..
A particle confined to a box cannot have an
arbitrary energy and also n=0 or E=0 is notadmissible
WAVE FUNCTIONS
The wave function of a particle in a box whoseenergy is E is
Substituting for E from equation (5) we get
)6.....(..........2
sin xmE
A
further )8.........(....................1
2
dx
-
7/27/2019 Schrodinger Equation
17/21
)10..(..........sin2
xL
n
Ln
)9.(..............................2
LA
gives
Therefore the normalized wavefunctions of theparticle are
The normalized wanefunctions
1,2, 3 are plotted as shown.
-
7/27/2019 Schrodinger Equation
18/21
Graphical explanation
A particle in a box with infinitely high
walls is an application of the Schrodingerequation which yields some insights intoparticle confinement. The wavefunctionmust be zero at the walls and the solution
for the wavefunction yields just sine waves.
The longest wavelength is = 2L andthe higher
modes have wavelengths given by
=2L/n where n= 1,2,3,.
-
7/27/2019 Schrodinger Equation
19/21
When this is substituted into the DeBroglierelationship it yields momentum p=h/
=nh/2LThe momentum expression for the particlein a box :
p=h/ =nh/2L: n= 1,2,3,..is used to calculate the energy associatedwith the particle
nEmL
hn
m
pmv
2
2222
822
1
-
7/27/2019 Schrodinger Equation
20/21
Though oversimplified, this indicates someimportant things about bound states for
particles:
1. The energies are quantized and can becharacterized by a quantum number n
2. The energy cannot be exactly zero.
3. The smaller the confinement, the largerthe energy required.
If ti l i fi d i t
-
7/27/2019 Schrodinger Equation
21/21
If a particle is confined into arectangular volume, the same kind ofprocess can be applied to a three-
dimensional "particle in a box", and thesame kind of energy contribution is madefrom each dimension. The energies for athree-dimensional box are
This gives a more physically realisticexpression for the available energies forcontained particles. This expression is usedin determining the density of possible
energy states for electrons in solids.
2
2
3
2
2
2
1
8mLE
nnnn
http://hyperphysics.phy-astr.gsu.edu/hbase/hyde.htmlhttp://hyperphysics.phy-astr.gsu.edu/hbase/davger.htmlhttp://hyperphysics.phy-astr.gsu.edu/hbase/hyde.htmlhttp://hyperphysics.phy-astr.gsu.edu/hbase/davger.htmlhttp://hyperphysics.phy-astr.gsu.edu/hbase/davger.htmlhttp://hyperphysics.phy-astr.gsu.edu/hbase/hyde.htmlhttp://hyperphysics.phy-astr.gsu.edu/hbase/davger.html