33 pius augustine matter waves and schrodinger equation

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Matter waves and Schrodinger equation (B Sc)

Light is a Wave;
It spreads out over Space.
Light is also a Particle that is at a Point”.
Interference, diffraction, polarization etc experiment establish
wave nature
Photoelectric effect, black body radiation etc establish particle
nature of light. (photon nature)
Conflicting ideas!!! But is.
Interference – super position of waves
Photoelectric effect – light packets called photons are needed.
Dr. Pius Augustine, SH College, Kochi

Wave is a disturbance produced in a medium. It
does not contain any particle or matter.
A wave is specified by the quantities like frequency
(ν), wavelength (λ), amplitude (A), intensity (I) and
phase velocity (v).
A particle contains some material and is specified by
its mass m, velocity v, momentum p and energy E.
Wave spreads – needs large space to occupy
Particle is confined– needs very small space.

Louis de Broglie (pronounced as duh Broyee)
had a question.
If radiation can exists in two forms – particle
and wave, why not matter?
His argument – Nature loves symmetry.
It lead to the discovery of matter waves and de
Broglie was awarded Nobel Prize in Physics in
1928.

Dual nature of matter or Wave particle duality
Friendship of contradictory aspects
Light shows dual nature
i. Wave nature – interference, diffraction,
polarization etc.
ii. Particle ( photon) – photoelectric effect
Nature loves symmetry.
Matter and radiation must have symmetrical
character.
ie. matter should also show dual nature.

For a photon, mc2 = hν = hc/λ
λ = h/p.
Louis de Broglie (1924) proposed that
wavelength λ associated with a
particle of momentum p (= mv)
λ = h/p.
Relation is called de Broglie relation
and wavelength of matter wave is
called de Broglie wavelength.

If an electron is accelerated from rest
through a p.d V,
eV = ½ mv2 = p2/2m
p = ( 2meV)1/2 = (2mE)1/2.
Sub value of constants, de Broglie eqn
λ = 12.27 Å
√V V – potential

Different expressions λ = h/p
λ = h/(2mE)1/2 E – kinetic energy
λ = h/(2mqV)1/2 q – charge of particle
(1/2 mv2 = qV)
When material particle like neutron is in thermal
equilibrium at temperature T, E =kT, where k –
Boltzmann’s constant (k = 1.38 x 10-23 J/K)
λ = h/(2mkT)1/2

Some explanation on λ = h/p
i)When v = 0, λ = h/0 = ∞
ie. matter waves are present only if the
particle is in motion.
ii) Higher the vel of the particle, shorter λ.
iii) Smaller the mass longer the λ
iv) de Broglie wavelength is independent of
charge of the body. Dr. Pius Augustine, SH College, Kochi

Some explanation on λ = h/p
v) Matter waves are not e-m waves. (in a medium, velocity
of e-m waves is constant, but matter waves velocity
depends on the velocity of the particle).
vi) If velocity of the particle is comparable with velocity of
light, mass will experience relativistic variation and bring
corresponding modification in λ.
vii) de Broglie wavelength is independent of nature of the
body.

de Broglie wavelength of an electron
Electron under a p.d of V volts.
½ mv2 = eV m and e – mass and charge of electron.
m2v2 = 2 meV or P = mv = (2meV)1/2
λ = h = h = h
P 2meV 2m E
λ = 12.27 Å
V
h = 6.62 x 10-34 Js
m = 9.1 x 10-31 kg
e = 1.602 x 10-19 C

Derive the expression λ = h/p
Momentum of photon p = hν/c = h/λ
So λ = h/p
Wavelength of the wave associated with material particle
λ = h/p = h/mv -------- (i)
Above expression can be derived from the equation
of a standing wave and principle of relativity

Derive the expression λ = h/p Continue….
Take an electron or proton – as a standing wave system in a region of
space occupied by the particle.
Let ψ be the quantity that undergoes periodic changes giving rise to
matter waves (function of time and space)
Ψ= ψo sinωt = ψo sin2πνt ---------- (ii)
ψo - amplitude of the wave at the point (x, y, z)
ν –frequency of the particle as observed by observer at rest relative to the
particle.

Derive the expression λ = h/p continue….
If the particle is moving with velocity v along +ve X- axis,
Ψ = ψo sin 2πν (t’+vx’/c2) ------- (iii)
(1-v2/c2)
From inverse Lorentz transformation theory of relativity,
t = (t’+vx’/c2)
(1-v2/c2)
Standard wave equation is
Ψ = ψo sin 2π (t’+x’/u’) -------- (iv)
T’
ψ Amplitude, T’ – periodic time, u’ phase velocity along X

Comparing eqn (iii) and (iv) 1/T’ = ν’ = ν
(1-v2/c2)
(t’+vx’/c2) = (t’+x’/u’)
u’ = c2/v
According to mass energy relation E = m0c2 = hν
ν = m0c2/h
ν’ = m0c2/h = mc2/h
(1-v2/c2)
Wavelength of the material particle = u’ = c2/v = h
ν’ mc2/h mv
Derive the expression λ = h/p Continue….

Note:
A material particle in motion involves two different velocities.
i) Refers to mechanical motion of the particle (v)
ii) Propagation of associated wave (u)
u = c2/v
A moving particle, whatever its nature has wave properties
associated with it, leads to a new form of mechanics called
wave mechanics, which replaces classical dynamics, when
fine structure details of the matter are to be considered.

Velocity of a body of mass 1kg is 1 m/s.
Calculate de Broglie wavelength? h = 6.62 x 10-34
Js. Ans: 4.85 x 10-11 m
Calculate the wavelength of an electron
accelerated under a p.d of 150V.
Hint: Ans: 1 Å
Calculate the energy of the neutron if its wavelength is 1
Å. Hint: 0.081 eV (1eV = 1.6 x 10-19 J)

An electron initially at rest is accelerated through a
potential difference of 5000 V. Compute the momentum,
de Broglie wavelength and the wave number of the
electron wave.
Hint: P = 3.82 x 10-23 kgm/s
λ = 0.173 Å
Wavenumber = 1/λ = 5.71 x 1010 per meter.

Pilot waves: de Broglie waves
de Broglie waves are not electromagnetic wave, a new
kind of waves (associated with quantnum material
particles) called pilot waves.
Pilot =guide the particle.
Matter waves consists of group of waves or a wave
packet.
Matter waves cannot be observed. It is a wave model
to describe and study matter.

Wave Packet Model : Phase velocity and Group Velocity
There is individual runner
velocity as well as group velocity
Wave Packet Model: In quantum physics, a
wave packet can be obtained by adding up
many waves having different wave numbers and
amplitudes.
The position of a particle can be localized within a specific region
by using such a wave packet which represents a particle, and it is
a group of many waves.
The velocity of the particle is equal to the velocity of the wave
packet.
In other words, the velocity of the particle is equal to the group of
the waves with which the wave packet is made of.

Phase velocity and Group Velocity
The velocity with which each individual wave travels is called phase
velocity or wave velocity.
Superposition of a number of waves having slightly different frequencies
form Group of waves.
The velocity of the group is different from the individual components of the
wave.
The velocity with which the resultant envelope of the group of waves
travels is called group velocity.
Phase velocity is greater than the group velocity in a normal medium

The velocity of the group or wave packet is called the
group velocity.
Monochromatic harmonic waves cannot carry
information through space.
In order to convey information, superposition of many
waves which gives a wave packet is needed.

Velocity of the point A – phase velocity, velocity of a point on the
envelope is group velocity k – wave vector
Phase velocity of A = ωk

• Group velocity: In an anomalous medium, the group velocity is
greater than the phase velocity.
• Phase velocity: In a normal medium, the phase velocity is
greater than the group velocity.
• Group velocity: The concept of group velocity is very
important when a particle is represented by waves.
• Phase velocity: The phase velocity is important when we deal
with individual waves.
• Group velocity: In a wave packet, the envelope travels with
the group velocity.
• Phase velocity: In a wave packet, the carrier wave travels with

Matter waves are not em waves. comment
Matter waves are not e-m waves. (in a medium, velocity of e-
m waves is constant, but matter waves velocity depends on
the velocity of the particle).
Phase velocity of matter waves depends on the wavelength,
even if the particle is moving in vacuum. But in vacuum all
em waves travel with the same velocity and is independent
of the wavelength.
EM waves are produced by accelerated charge particle.

Distinguish between light waves and matter waves?
EM vs Matter waves.
A proton and an electron have same de Broglie
wavelength. Which possess more K.E?
mE = h2/2λ2.
since λ is same for both,
mE = constant.
higher mass lower E and
vice. mp>me.

Matter waves are called probability waves. Comment
The amplitude of matter wave speaks about the
probability of finding a particle in space at a
particular instant. A large wave amplitude means a
large probability of finding the particle at that
position.

Wave particle duality – further thought
Propagation experiment: interference, diffraction etc
wave nature of light established
Interaction experiment: interaction between radiation
and matter, particle nature of light.
Electron – group of matter waves
Photon – group of em waves

Wave particle duality – further thought
There are only waves- in groups, it is wave
packets which represent particles.
Interaction between objects is nothing but
interaction between the wave groups associated
with the particles.

When an electron wave strikes a boundary, reflected
or refracted part of the wave represent electron?
According to de Broglie, particles are real, but wave is
abstract quantity.
Matter waves are probability waves – amplitude tells the
probability of finding a particle in space at a particular
instant.
It is a model. If only one electron, it will be either reflected or
refracted. But, if there are number of particles (electrons),
then partly reflection and partly refraction.

Why a fast neutron beam needs to be thermalised with
the environment, before it can be used for neutron
diffraction experiments?
For diffraction to take place wavelength of the particle
used should be comparable with the interatomic
distance.
The wave length of fast neutrons is very small. But when
they are slowed down their energy is reduced and they
become thermal neutron. Their wavelength becomes
comparable with interatomic spacing.

Note: a model need not give complete description.
Two models i) Wave model ii) particle model.
For complete description
It is wave and particle model for electron
Not wave or particle model for electron.
Two models complement each other and together
give a description of matter.

Is the de Broglie wavelength of a photon of an em
radiation equal to the wavelength of the radiation?
Yes. de Broglie wavelength λ = h/p
For a photon p = hν/c
λ = h/p = c/ν. Two wavelengths are the same
State one application of electron wave?
Electron microscope
Finding the size of the nucleus, quarks etc, where wavelength of
electron wave can be modulated by changing the speed of
electron for diffraction

An electron and a proton are possessing same
amount of kinetic energy, which of the two has
greater de Broglie wavelength? Justify your
answer.
For const E, λ α 1/√m
λ will be greater for smaller mass
Show that dimensional formula for h/p is [L]
Are matter waves em?

Davisson and Germer’s Experiment
First experimental evidence showed that beam of material particles
exhibit wavelike properties
Experimental verification of Matter waves. (diffraction of electron waves)
Succeeded in measuring the de
Broglie wavelengths for slow
electrons accelerated by a
potential difference ranging from
30 to 100 V by diffraction
methods

Electron waves were used ??
1.Intense beam can be easily produced
2.Velocity can be easily controlled
3.Wavelength of electron beam (waves) is of the same
order as that of X-rays – because mass of electrons is
small.
Note: optical grating cannot be used as λ is much
smaller that visible light (crystal grating–matching λ)Dr. Pius Augustine, SH College, Kochi

Electron gun: creating a
focused (using slits)
strong beam of electrons.
(accelerating potential
controls the velocity of
electrons.
Electron beam will be directed into a high vacuum to fall at an
angle on a large single crystal of Nickel target.
Electrons reflected from the crystal are collected by a rotating
Faraday cylinder called collector (C), which is connected to a
sensitive galvanometer.
Tungsten filament heated to red. – thermionic emission

Evacuated chamber for the free movement of electron without any air
resistance
Electron gun for the emission of electron
Battery Used for the acceleration of electrons inside the cylinder
Cylinder with fine hole along its axis connected to the battery so that
electrons entering it could be accelerated to high speed
Nickel Target used to deflect electron beam towards the detector
Movable detector (collector) to detect the intensity and scattering of electrons
deflected by the nickel crystal at varying voltage supplies(44 to 68 V)
Galvanometer to measure the small values of current

Davisson Germer electron diffraction
Experiment
Focussed and accelerated electron beam from an
electron gun is directed in a high vacuum to fall
at an angle on a large single crystal of Ni target.
Angular distribution of reflected electrons is
studied with a rotating Faraday cylinder (on a
circular scale 20o to 90o) connected to a
sensitive galvanometer.

Faraday cylinder is surrounded by a protecting
cylinder with retarding potential (= 9/10th
accelerating pot).
Only fastest electrons possessing nearly incident
velocity (not secondary electrons produced by
collision) would reach.
Succeeded in measuring de Broglie wavelengths
for slow electrons accelerated by a potential of
30 to 100V. Dr. Pius Augustine, SH College, Kochi

Experiment is done for different accelerating
potential by moving Faraday cylinder.
Galvanometer current (measure of intensity of
diffracted electrons) is plotted against
colatitude Φ (angle between incident beam and
beam entering cylinder)

At higher potential and lower potential bump disappears.
The current which is a measure of intensity of the diffracted beam, is
plotted against φ.
Φ is the angle between incident beam direction and the direction in which
the beam entering the cylinder.
Reference direction is the direction of incident beam (marked 0o)
Value of intensity marked at different angles of collector
Only Y axis – which is intensity.
For a particular voltage – one graph is plotted by noting the intensity
values at different angles (φ). It is a polar graph.
φ= 30,35,40,45
Intensity→
Incident
direction
Φ = 50o
Φ = 50o
Φ = 50o
Φ = 50o

Observations:
Strong peak was detected at 55V, and the angle of
scattering was observed to be 50°
The pattern of deflected electrons was quite similar
to the diffraction pattern of waves
The wavelength corresponding to the electron (matter
wave) was found to be λ = 0.167nm
The experiment was in strong agreement with De
Broglie’s hypothesis

A bump begins to appear in the curve
for 44 V electrons.
Bump becomes most prominent in the
curve for 54 volts electrons at Φ =50o.
This verifies diffraction of electrons.
λ = 12.27 Å
√V
λ = 12.27 Å = 1.67 Å
√54 Dr. Pius Augustine, SH College, Kochi

X- ray diffraction has already shown that Ni
crystal acts as a plane diffraction grating with
grating space
d = 0.91Å
First order diffraction using Bragg eqn is
λ = 2dsinθ = 1.65Å
(excellent agreement with λ computed from de
Broglie hypothesis)

When scattering angle φ = 50o
the corresponding angle of
incidence relative to the family of
Bragg planes is (glancing angle)
θ = 65o. (ie. glancing angle = 90
-25)

Why don’t we observe wave nature
in daily life ?
For macroscopic object ( say a bullet m
= 1g, moving with velocity 103m/s)
λ = h/mv = 6.625 x 10-34m.
This wavelength is very small to show
effect.

G. P. Thomson’s Experiment
George Paget Thomson
1892 - 1975
Established wave nature of high energy
electrons.
Thermionic emission of electrons
Accelerated by high p.d inside a discharge tube.
Electron pass through a fine hole in a metal block and falls on gold
foil.
Scattered electrons will strike the photographic plate.

Au foil consists of very large number of microscopic crystals
oriented at random.
Some of them will be at corrent angle to give Bragg’s diffraction.
nλ = 2dsinθ
On processing the photographic plate, central spot (undeflected
electrons) surrounded by a series of concentric diffraction rings
were obtained.
G. P. Thomson’s Experiment - continue….

Ist order diffraction, from Bragg’s law d = λ/2sinθ = λ/2θ (θ
– is small)
From fig: 2θ = r/L d = λ (L/r) ------- (i)
de Broglie eqn. λ = h/mv, m – relativistic mass of electrons
v-vel.
Note: P.D applied was (15 -60) kV. So, relativistic variation.Dr. Pius Augustine, SH College, Kochi

de Broglie eqn. λ = h/mv
Relativistic expression for KE = mc2 – moc2. = eV
eV – energy acquired under p.d of V volts.
----- (ii)
------- (iii)
Z = eV/moc2

Multiplying eqn (ii) and (iii)
Z = eV/moc2
Substitute in λ = h/mv

Z = eV/moc2
Relativistic expression for de Broglie wavelength of
an electron accelerated through a high p.d of V volts.Dr. Pius Augustine, SH College, Kochi

If relativistic effect is ignored (v/c << 1), the above expression reduces to
Sub λ in eqn for d
Comparison of d obtained in this way agreed to within 1%
with values determined using X-rays of known wavelength.
In the case of gold foil, 4.08 Å and X –rays it was 4.06 Å

de Broglie wavelength associated with electron accelerated
through a potential difference
Non relativistic
λ = h/p p – momentum h = Planck’s const = 6.62 x 10-34 Js
Kinetic energy acquired = EK = eV = ½ mv2.
p = mv = (2mEK)1/2.
Mass of electron = 9.1 x 10-31 kg,
Charge of electron = 1.6 x 10-19 C

Relativistic (v is comparable with c) EK = eV but ≠ ½ mv2.
λ = h/p p – momentum h = Planck’s const = 6.62 x 10-34 Js
E and p are related as E2 = p2c2 + m0
2c4.

Relativistic (v is comparable with c) EK = eV but ≠ ½ mv2.
λ = h/p

Calculate the de Broglie wavelength of neutron of
energy 28.8 eV?
Hint: m = 1.67 x 10-27 kg. λ = h/(2mE)1/2. Ans: 4.2 Å
Find the energy of neutron in units of electron volt
whose de Broglie wavelength is 1 Å?
Hint: m = 1.67 x 10-27 kg. λ = h/(2mE)1/2.
1 eV = 1.602 x 10-19 J Ans: 0.0813 eV
Calculate the de Broglie wavelength associated with a
proton moving with velocity = 1/20th of c?
Hint: m = 1.67 x 10-27 kg. Ans: 2.634 x 10-14 m

What is the energy of a gamma ray photon having wavelength
= 1 Å.
Hint: E = h ν = hc/λ 1 Å = 10-10 m Ans: 1.24 x 104 eV
What voltage must be applied to an electron microscope to
produce electrons of wavelength 0.50 Å.
Hint: m = 9.0 x 10-31 kg. λ = h/(2meV)1/2.
e= 1.602 x 10-19 C Ans: 602.4 volts
Calculate the wavelength of thermal neutrons at 27 oC,
assuming energy of a particle at absolute temperature T is of
the order of kT, k-Boltzmann’s constant= 1.38 x 10-23 J/K
Hint: m = 1.67 x 10-27 kg. λ = h/(2mE)1/2. λ = h/(2mkT)1/2. T =
300 K
Ans: 1.77 Å

Calculate the de Broglie wavelength of an alpha particle
accelerated through a potential difference of 2000V. Given
mass of proton = 1.67 x 10-27 Kg and h = 6.62 x 10-34 Js.
Hint: λ = h/(2mE)1/2 = h/(2mqV)1/2 q = 2 e, m = 4 x mp
Ans: 2.3 x 104 Å
Show that the de Broglie wavelength for a material particle of
rest mass m0 and charge e, accelerated from rest through a
P.D of V volts relativistically is
A beam of monoenergetic neutrons corresponding to 207 oC is
allowed to fall on a crystal. A first order reflection is
observed at a glancing angle 300; calculate the interplanar
spacing of the crystal.
Hint: 1* h/(2mkT)1/2 = 2d sindθ T = 300K
Ans: 1.78 ÅDr. Pius Augustine, SH College, Kochi

de Broglie concept of stationary orbit
Bohr atomic theory: “only those electron orbit are allowed as
stationary orbits in which the angular momentum of electron is
an integral multiple of h/2π”. J = nh/2π, n = 1, 2, 3 …
It was just an assumption, and it worked.
According to de Broglie, if wave properties are associated with electron,
then a sort of resonance might occur, when the circumference of the
orbit is equal to an integral multiple of the wavelength of the wave
associated with electron.

de Broglie concept of stationary orbit
Stationary orbits – circumferrence = n (wavelength of
electron)
2πr = nλ, r – radius of the orbit,
and λ – de Broglie wavelength of electron
2πr = nh/mv
mvr = nh/2π - which is Bohr’s postulate

Phase velocity and wave velocity - Recollect
When a single wave of definite wavelength travels in a medium,
its velocity of propagation in the medium is called “wave
velocity” or “phase velocity”.
When number of waves of different wavelengths are moving with
different velocities in a medium, then the observed velocity is
the velocity of the wave packet (or wave group) formed by the
waves. This is group velocity which is less than the wave
velocity.

Obtain expression for group velocity.
Two SH waves – slightly different λ and
wave velocities.
When two waves are in phase, resultant gives
maximum amplitude and vice versa.
At a given instant maximum will be formed at say point P.
At a later instant the maximum will be formed little to the
left of P, and then further left etc.
ie. maximum of the group moves with a velocity slightly less
than that of the individual waves.

Consider two waves having same amplitude ‘a’
y1 = a sin(ωt-kx) and y2 = a sin[(ω+dω)t-(k+dk)x]
k = 2π/λ is propagation constant
Resultant displacement y = y1 + y2.
= a sin(ωt-kx) + a sin[(ω+dω)t-(k+dk)x]
Using sin A + sin B = 2 sin [(A+B)/2]cos[(A-B)/2]
y = y1 + y2.
y = 2a sin[(ω+dω/2)t - (k+dk/2)x]cos (dω)t-(dk)x
2
= 2a sin[(ω)t - (k)x]cos (dω)t-(dk)x
2
Neglecting
dω/2 and dk/2

y = 2a cos (dω)t - (dk)x sin[(ω)t - (k)x]
2 2
Two parts sine part and cosine part.
Sine factor represents wave of same frequency ν and same
propagation constant k, as that of the component waves
having velocity u = ω/k
Cosine factor – slowly varying function of both x and t and
modulates the amplitude of the resultant wave. It represents
the wave group whose (group) velocity is vg = dω/dk
Group velocity Vg = dω/dk

Show that group velocity (vg) < phase (wave) velocity (u)
d(2π/λ) = -1/λ2 * dλ
Subtracted term
So, vg< u
If dispersion term du/dλ = 0, vg = u

Requirement of wave packet??
de Broglie theory, λ = h/mv, m – mass of the particle,
v-velocity of the moving particle.
Energy of a wave E = hν = mc2. wave energy = particle energy
Frequency, ν = mc2/h wavelength, λ = h/mv
de Broglie wave velocity or phase velocity u = νλ
u = (mc2/h) (h/mv) = c2/v
v < c ie. c2/v > c or u > c
But, special theory of relativity, c is highest velocity.
Here, wave velocity is greater than velocity of light ???

Requirement of wave packet??
Also, this equation implies that, wave associated with a
moving particle would travel faster than particle itself??
ie. particle will be far behind!!! ???
It ruled out that, material particle in motion cannot be
represented by a single wave train..
????
Schrodinger solved the puzzle by stating that material
particle in motion is equivalent to a wave packet, not
single wave. Dr. Pius Augustine, SH College, Kochi

Wave packet: Note::
Group of waves, Slightly different velocity and wavelength
Phases and amplitudes are so chosen that they interfere
constructively over only a small region of space, where the
particle can be located – probability maximum.
Outside – destructive interference- amplitude reduces to zero.
Velocity is group velocity = vg.
Average velocity of the individual wave is phase velocity (u).
Velocity of material particle (v)=Group velocity (vg)

Show that: Velocity of material particle (v)=Group velocity (vg)
de Broglie wave velocity or phase velocity u = νλ
Frequency is related to angular frequency: ω = 2πν
ω/u = 2π/λ
k = 2π/λ
K.E of the particle = Total energy – Potential energy
½ mv2 = E -V

Show that: Velocity of material particle (v)=Group velocity (vg)
1 = mv = m 2 (E-V)
λ h h m
Velocity of material particle (v) = Group velocity (vg)Dr. Pius Augustine, SH College, Kochi

Wave packet + guiding wave = particle moving with velocity
Material particle in motion is equivalent to a group of waves or a
wave packet.
Wave packet will be soon dissipated. (compare with water wave)
So, the requirement of a quiding wave. (compare with carrier wave
in propagation).
Postulated the existence of a guiding wave which is described by
Schrodinger equation.

Physical significance of Schrodinger equation
Schrodinger equation relates the amplitude of the guiding wave to
the probability of finding a material particle at a point.
If the amplitude of the guiding wave is zero at a certain pint in space,
the probability of finding the material particle at that point is
infinitesimal.
Mechanical process is accompanied by a wave, the square of the
amplitude gives a measure of the probability of the event taking place
at the point.

Note: Value of amplitude is very small except in the
wave packet.
Probability of finding the electron within the region
of the packet is maximum.
(Wave Packet + Guiding Wave ) possess the
properties of a particle moving with velocity v and
exhibits interference, diffraction phenomenon.Dr. Pius Augustine, SH College, Kochi

Relation between phase velocity (u) and group velocity (vg) for a
non relativistic particle
u = νλ ν – frequency of the wave, λ - wavelength of the wave
λ = h/mv v – velocity of the material particle.
For a free particle, P.E = 0, T. E = K. E = ½ mv2.
E = hν ν = E/h = ½ mv2/h
u = νλ = ½ (mv2/h) (h/mv) = v/2 = vg/2
For a non relativistic free particle phase velocity is half the
group velocity.
ie. vg > u (for non relativistic free particle)
What do you understand from this??? Dr. Pius Augustine, SH College, Kochi

Schrodinger Equations:
Equation of motion of Matter waves
3 cases
1. Time dependent Schrodinger equation
2. Schrodinger equation for a free particle
3. Time independent Schrodinger equation

The Schrödinger equation (also known as
Schrödinger’s wave equation) is a partial differential
equation
It describes the dynamics of quantum mechanical
systems via the wave function.
The trajectory, the positioning, and the energy of these
systems can be retrieved by solving the Schrödinger
equation. Dr. Pius Augustine, SH College, Kochi

Recollect from EM theory - Maxwell’s equations
c – Speed of light in vacuum
E – Electric field
B – Magnetic field
ρ - Charge density
μ0 – Magnetic permeability of free space
ε0 - electrical permittivity of free space
J - Current density

Recollect from EM theory - Maxwell’s equations
The first equation is the basis
of electric generators,
inductors, and transformers.
Faraday’s Law
Fourth equation says that no
magnetic monopoles will
exist.

Recollect from EM theory – Wave equations
This is electro magnetic wave equation in 3-dimensions.
This wave equation is applicable not only in electro
magnetics– but also in acoustics, seismic waves, sound
waves, water waves, and fluid dynamics.

All of the information for a subatomic particle is encoded
within a wave function.
This wave function will satisfy Schrodinger equation,
which can be solved.
Schrodinger’s original approach is presented in the following
slides

Time dependent Schrödinger equation
Guiding wave obeying Schrodinger equation is the essential requirement
to have a non dissipating wave packet.
Stationary waves associated with a moving particle.
ψ(r,t) – displacement for the de Broglie waves at any location r
= xi + yj + zk at time t.
Diiferential equation of the wave motion in 1D (according to
Maxwell’s wave equation) is

Time dependent Schrödinger equation – continue…
It is second order partial differential equation and is satisfied with plane
wave solutions
k = 2π/λ and ω = 2πf.
Energy of a photon E = hν = ℏω
de Broglie equation p = h/λ = ℏk
h/2π =ℏ
This is plane wave equation describing a photon
So, wave equation can be
E2 = p2c2 + m2c4 (for photon m = 0)

Time dependent Schrödinger equation continue..
But in wave mechanics, we deal with relativistic particles
So, E2 = p2c2 + m2c4 Also, we are dealing with stationary
waves associated with a moving particle.
E is replaced with ψ(r,t) – displacement for the de Broglie
waves at any location r = xi + yj + zk at time t.

Time dependent Schrodinger equation continue..
Now small trick Back solve for an operator to get the previous equation
This equation, if rearranged, it is Klein Gordon equation
In 3D, this can be written as
Klein Gordon equation for a free particleDr. Pius Augustine, SH College, Kochi

A few manipulations in the
form(1+x)1/2
Tayor series
= first two terms for small x
Tayor series
(p = mv) << mc

This energy is substituted in the wave equation

Take first and second derivatives
Last term is neglected, because of c2 in the exponential term
Substitute in the starting equation Dr. Pius Augustine, SH College, Kochi

This equation, if rearranged, it is Klein Gordon equation
In 3D, this can be written as
1D

Compare with classical Hamiltonian - the term on the right-hand
side of the equation describes the total energy of the wave
function. (We did the derivation for free particle)
Schrödinger Equation in three dimensions with a potential is
given by:
EH
H ψ = E ψ
H operates on ψ = eigen value (E) ψ

Time independent Schrödinger equation
For de-Broglie wave associated with material particle, c is
replaced with phase (wave) velocity u, and E by ψ
Solution gives ψ as a periodic
displacement of time
Ψ(r,t) = ψ0(r) e-iωt

Time independent Schrödinger equation continue…..
Differentiating twice w.r.to time
Substitute in the above equation
ω = 2πν
= 2πu/λ
ω/u = 2π/ λ
ω2/u2 = 4π2/ λ2

Time independent Schrödinger equation continue…
Now introduce wave mechanical concept
ie. include de Broglie eqn λ = h/mv
So wave equation in the previous slide becomes
K.E = ½ mv2 = Tot E (E) - P.E (V)
m2v2 = 2m (E-V)
h/2π =ℏ
V = 0 for free particle
Time independent Schrodinger equation T I Sch eqn for free particle

Time dependent Schrödinger equation (alternate method)
Differentiating w.r.to time
= 2π i (E/h) ψ

Time dependent Schrödinger equation (alternate method)
Equation contains time. It is Time dependent Sch eqn.
Operator H = is called Hamiltonian
Hψ = Eψ

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33 pius augustine matter waves and schrodinger equation

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