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Physics-I
Dr. Anurag Srivastava
Web address: http://tiiciiitm.com/profanurag
Email: [email protected]
Visit me: Room-110, Block-E, IIITM Campus
2
• Electrodynamics: Maxwell’s equations: differential and integral forms, significance of Maxwell’s equations, displacement current and correction in Ampere’s law, electromagnetic wave propagation, transverse nature of EM waves, wave propagation in bounded system, applications.
• Quantum Physics: Dual nature of matter, de-Broglie Hypothesis, Heisenberg uncertainty principle and its applications, postulates of quantum mechanics, wave function & its physical significance, probability density, Schrodinger’s wave equation, eigen values & eigen functions, Applications of Schroedinger equation.
Syllabus
Erwin Rudolf Josef Alexander Schrödinger
Born: 12 Aug 1887 in Erdberg, Vienna, Austria
Died: 4 Jan 1961 in Vienna, Austria
Nobel Prize in Physics 1933
"for the discovery of new productive forms of atomic theory"
An equation for matter waves
2
2Try
t x
2 2
2i
2t m x
Seem to need an equation
that involves the first
derivative in time, but the
second derivative in space ( , ) is "wave function" associated with matter wavex t
,
i kx tx t e
As before try solution
So equation for matter waves in free space is
(free particle Schrödinger equation)
• Solve this equation to obtain y
• Tells us how y evolves or behaves
in a given potential
• Analogue of Newton‟s equation
in classical mechanics
Schrödinger’s Equation
Erwin Schrödinger (1887-1961)
• This was a plausibility argument, not a derivation. We believe the Schrödinger equation not because of this argument, but because its predictions agree with experiment.
• There are limits to its validity. In this form it applies only to a single, non-relativistic particle (i.e. one with non-zero rest mass and speed much less than c)
• The Schrödinger equation is a partial differential equation in x and t (like classical wave equation). Unlike the classical wave equation it is first order in time.
• The Schrödinger equation contains the complex number i. Therefore its solutions are essentially complex (unlike classical waves, where the use of complex numbers is just a mathematical convenience).
Schrödinger’s Equation
Interpretation of |y|2
• The quantity |y|2 is interpreted as the probability that the
particle can be found at a particular point x and a
particular time t
• The act of measurement „collapses‟ the wave function
and turns it into a particle
Neils Bohr (1885-1962)
An equation for matter waves
2
( , )2
pE V x t
m
2 2
2i ( , )
2V x t
t m x
For particle in a potential V(x,t)
Suggests modification to Schrödinger equation:
Time-dependent Schrödinger equation
Total energy = KE + PE
Schrödinger
What about particles that are not free?
2 2
, ,2
kx t x t
my y
Has form (Total Energy)*(wavefunction) = (KE)*(wavefunction)
2 2
2i
2t m x
,i kx t
x t e
Substitute into free particle equation
2
2
pE
m
(Total Energy)*(wavefunction) = (KE+PE)*(wavefunction)
gives
The Schrödinger equation: notes
2 2
2i ( , )
2V x t
t m x
• This was a plausibility argument, not a derivation. We believe the Schrödinger
equation not because of this argument, but because its predictions agree with
experiment.
• There are limits to its validity. In this form it applies only to a single, non-
relativistic particle (i.e. one with non-zero rest mass and speed much less than c)
• The Schrödinger equation is a partial differential equation in x and t (like
classical wave equation). Unlike the classical wave equation it is first order in time.
• The Schrödinger equation contains the complex number i. Therefore its
solutions are essentially complex (unlike classical waves, where the use of
complex numbers is just a mathematical convenience).
• Note the +ve sign of i in the Schrödinger equation. This came from our looking
for plane waves of the form
We could equally well have looked for solutions of the form
Then we would have got a –ve sign.
This is a matter of convention (now very well established).
i te i te
The Hamiltonian operator
2 2
2i ( , )
2V x t
t m x
ˆi Ht
Can think of the RHS of the Schrödinger equation as a
differential operator that represents the energy of the
particle.
Hence there is an alternative
(shorthand) form for the time-
dependent Schrödinger equation:
This operator is called the Hamiltonian of
the particle, and usually given the symbol H2 2
2ˆ( , )
2
dV x t H
m dx
Kinetic
energy
operator
Potential
energy
operator
Hamiltonian is a linear differential operator.
Schrödinger equation is a linear homogeneous partial differential equation
Time-dependent
Schrödinger equation
Interpretation of the wave function
2 2
0( , ) ( , ) d
bb
xx a a
x t x x t x
2 * 2
( , )x t x
Ψ is a complex quantity, so how can it correspond to real physical
measurements on a system?
Remember photons: number of photons per unit volume is proportional to
the electromagnetic energy per unit volume, hence to square of
electromagnetic field strength.
Postulate (Born interpretation): probability of finding particle in a small
length δx at position x and time t is equal to
Note: |Ψ(x,t)|2 is the probability per unit length. It is
real as required for a probability distribution.
Total probability of finding particle
between positions a and b is
a b
|Ψ|2
x
δx
Born
An equation for matter waves:
the time-dependent Schrödinger equation
2 2
2 2 2
1
x v t
wave velocityv
Classical 1D wave
equation
e.g. waves on a string:
Can we use this to describe matter waves in free space?
( , )x t
x ( , ) wave displacementx t
Try solution ,
i kx tx t e
But this isn‟t correct! For free particles we know that
2
2
pE
m
DOUBLE-SLIT EXPERIMENT REVISITED
Detecting
screen
Incoming coherent
beam of particles
(or light)
sind
D
θ
y
1
2
Schrödinger equation is linear: solution with both slits open is 1 2
Observation is nonlinear 1 2
2 2 2 * *
1 2 2 1
Usual “particle” part Interference term
gives fringes
Normalization
Total probability for particle to be somewhere should always be one
Suppose we have a solution to the
Schrödinger equation that is not
normalized. Then we can
•Calculate the normalization integral
•Re-scale the wave function as
(This works because any solution to
the SE multiplied by a constant
remains a solution, because the SE
is LINEAR and HOMOGENEOUS)
Normalization condition 2( , ) d 1x t x
1( , ) ( , )x t x t
Ny y
2( , ) dN x t x
New wavefunction is normalized to 1
A wavefunction which obeys this
condition is said to be normalized
2 2( , ) ,
( , ) 0,
x t a x a x a
x t x a
Particle with un-normalized wavefunction
at some instant of time t
Normalizing a wavefunction - example
Conservation of probability
Total probability for particle to be somewhere should ALWAYS be one
2( , ) d constantx t x
If the Born interpretation of the wavefunction is correct then the normalization
integral must be independent of time (and can always be chosen to be 1 by
normalizing the wavefunction)
We can prove that this is true for physically relevant wavefunctions
using the Schrödinger equation. This is a very important check on the
consistency of the Born interpretation.
2 2
2i ( , )
2V x t
t m x
2( , ) d constantx t x
Boundary conditions for the wavefunction
The wavefunction must:
1. Be a continuous and single-valued
function of both x and t (in order that
the probability density is uniquely
defined)
Examples of unsuitable wavefunctions
Not single valued
Discontinuous
Gradient discontinuous
x
( )xy
( )xy
x
( )xy
x
2. Have a continuous first derivative
(except at points where the potential is infinite)
3. Have a finite normalization integral
(so we can define a normalized probability)
Time-independent Schrödinger equation
Suppose potential is independent of
time
2 2
2i ( )
2V x
t m x
LHS involves only
variation of Ψ with t
RHS involves only variation of Ψ
with x (i.e. Hamiltonian operator
does not depend on t)
Look for a separated solution ( , ) ( ) ( )x t x T ty
Substitute:
, ( )V x t V x
2 2
2( ) ( ) ( ) ( ) ( ) ( ) ( )
2x T t V x x T t i x T t
m x ty y y
2 2
2 2( ) ( ) ( )
dx T t T t
x dx
yy
2 2
2( )
2
d dTT V x T i
m dx dt
yy y
N.B. Total not partial
derivatives now
etc
2 2
2
1 1( )
2
d dTV x i
m dx T dt
y
y
Divide by ψT
LHS depends only on x, RHS depends only on t.
True for all x and t so both sides must be a constant, A (A = separation constant)
2 2
2( )
2
d dTT V x T i
m dx dt
yy y
1 dTi A
T dt
2 2
2
1( )
2
dV x A
m dx
y
y
So we have two equations, one for the time dependence of the wavefunction and
one for the space dependence. We also have to determine the separation constant.
This gives
/( ) iEtT t aeThis is like a wave i te
with /A . So A E .
1 dTi A
T dt
dT iAT
dt
/( ) iAtT t ae
1 dTi A
T dt
2 2
2
1( )
2
dV x A
m dx
y
y
SOLVING THE TIME EQUATION
• This only tells us that T(t) depends on the energy E. It doesn‟t tell us what the
energy actually is. For that we have to solve the space part.
• T(t) does not depend explicitly on the potential V(x). But there is an implicit
dependence because the potential affects the possible values for the energy E.
21
TIME INDEPENDENT SCHROEDINGER EQUATION
Erwin Schroedinger
Consider a particle of mass „m‟, moving with a velocity
„v‟ along + ve X-axis. Then the according to de Broglie
Hypothesis, the wave length of the wave associated
with the particle is given by
mv
h
A wave traveling along x-axis can be represented by the equation
xktieAtx ,
22
Where Ψ(x,t) is called wave function. The differential equation of matter wave in one
dimension is derived as
08
2
2
2
2
yy
VEh
m
dx
d
The above equation is called one-dimensional Schroedinger’s wave equation in one
dimension.In three dimensions the Schroedinger wave equation becomes
08
08
2
22
2
2
2
2
2
2
2
2
y
y
yyyy
VEh
m
VEh
m
zyx
2 2
2
d( )
2 dV x E
m x
yy y
With A = E, the space equation becomes:
This is the time-independent Schrödinger
equation
H Ey y
Solving the space equation = rest of course!
Time-independent Schrödinger equation
or
But probability
distribution is static
2 * / /
2*
, , ( ) ( )
( ) ( ) ( )
iEt iEtP x t x t x e x e
x x x
y y y
y y y
/( , ) ( ) ( ) ( ) iEtx t x T t x ey y Solution to full TDSE is
Even though the potential is independent of time the wavefunction still oscillates in time
For this reason a solution of the TISE is known as a stationary state
Notes
• In one space dimension, the time-independent Schrödinger equation is an ordinary differential equation (not a partial differential equation)
• The time-independent Schrödinger equation is an
eigenvalue equation for the Hamiltonian operator:
Operator × function = number × function
(Compare Matrix × vector = number × vector)
• We will consistently use uppercase Ψ(x,t) for the full wavefunction (TDSE), and lowercase ψ(x) for the spatial part of the wavefunction when time and space have been separated (TISE)
H Ey y
2 2
2
d( )
2 dV x E
m x
yy y
Energy of the electron
Energy is related to the Principle Quantum number, n.
This gives 3 of the 4 quantum numbers, the last one is the spin
quantum number, s, either +½ or – ½.
Wave
Functions: Probability to find
an electron
Energy of the electron
Electron Transitions give off Energy as
Light/Xrays
E=hc/
Red/Blue Clouds in Space
Zeeman Effect
Light Emission in Magnetic Field
Multi-electron Atoms
(Wolfgang) Pauli Principle Exclusion Principle
No 2 electrons with same quantum numbers!
Energy Eigenvalues
Eigen value and Eigen Function
Applications:
Operators in Quantum Mechanics