Transcript Schrödinger Equation

Photoelectric Effect
(How Einstein really became
famous!)
Photoelectric Effect
Metal Foil
Photoelectric Effect
Metal Foil
Photoelectric Effect
• As blue light strikes the metal foil,
the foil emits electrons.
Photoelectric Effect
Photoelectric Effect
• When red light hits the metal foil,
the foil does not emit electrons.
• Blue light has more energy than
red light.
• How could we get more energy
into the red light?
• Try increasing the brightness.
Photoelectric Effect
Photoelectric Effect
• Well, that didn’t work!
• Maybe its still not bright enough.
Photoelectric Effect
Photoelectric Effect
• Still not working.
• What happens with brighter blue
light?
Photoelectric Effect
Photoelectric Effect
• More blue light means more
electrons emitted, but that doesn’t
work with red.
Photoelectric Effect
Photoelectric Effect
• Wave theory cannot explain
these phenomena, as the energy
depends on the intensity
(brightness)
• According to wave theory bright
red light should work!
►BUT
IT DOESN’T!
Photoelectric Effect
• Einstein said that light travels in
tiny packets called quanta.
• The energy of each quanta is
given by its frequency frequency
E=hf
Energy
Planck’s constant
Photoelectric Effect
• Each metal has a minimum energy
needed for an electron to be emitted.
• This is known as the work function, W.
• So, for an electron to be emitted, the
energy of the photon, hf, must be
greater than the work function, W.
• The excess energy is the kinetic energy,
E of the emitted electron.
Most commonly observed phenomena with light can be explained by waves.
But the photoelectric effect suggested a particle nature for light.
Photoelectric Effect
EINSTEIN’S PHOTOELECTRIC EQUATION:-
Blackbody Radiation
Contents
• Definition of a Black-Body
• Black-Body Radation Laws
*1- The Rayleigh-Jeans Law
2- The Wien Displacement Law
3- The Stefan-Boltzmann Law
*4- The Planck Law
• Application for Black Body
• Conclusion
• Summary
Motivation
• The black body is importance in
thermal radiation theory and practice.
• The ideal black body notion is
importance
in studying thermal
radiation and electromagnetic
radiation transfer in all wavelength
bands.
• The black body is used as a standard
with which the absorption of real
bodies is compared.
Definition of a black body
A black body is an ideal body
which allows the whole of the
incident radiation to pass into
itself ( without reflecting the
energy ) and absorbs within itself
this whole incident radiation
(without passing on the energy).
This propety is valid for radiation
corresponding to all wavelengths
and to all angels of incidence.
Therefore, the black body is an
ideal absorber of incident
radaition.
Black-Body Radiation Laws
I ( , T ) 
The Rayleigh-Jeans Law.
* It agrees with experimental
measurements for long
wavelengths.
* It predicts an energy output
that diverges towards
infinity as wavelengths
grow smaller.
* The failure has become
known as the ultraviolet
catastrophe.
2ckT
4
I ( , T ) 
2ckT
Ultraviolet Catastrophe
4
• This formula also had a
problem. The problem
was the
term in the
denominator.
• For large wavelengths it
fitted the experimental
data but it had major
problems at shorter
wavelengths.
Black-Body Radiation Laws
Planck Law
- We have two forms. As a
function of wavelength.
2hc 2
I ( , T ) 
5
1
hc
e k T
1
And as a function of frequency
I ( , T ) 
2 h
c2
3
1
h
e kT 1
The Planck Law gives a
distribution that peaks at a
certain wavelength, the peak
shifts to shorter wavelengths
for higher temperatures, and
the area under the curve
grows rapidly with increasing
temperature.
Comparison between Classical and Quantum
viewpoint
There is a good fit at long wavelengths, but at short wavlengths there is a
major disagreement. Rayleigh-Jeans
∞, but Black-body
0.
http://upload.wikimedia.org/wikipedia/commons/a/a1/Blackbody-lg.png
Radiation Curves
Conclusion
• As the temperature
increases, the peak
wavelength emitted by
the black body
decreases.
• As temperature
increases, the total
energy emitted
increases, because the
total area under the
curve increases.
• The curve gets infinitely
close to the x-axis but
never touches it.
The Birth of Quantum Mechanics
___________________________
• At the turn of the last century, there were
several experimental observations which could
not be explained by the established laws of
classical physics and called for a radically
different way of thinking
• This led to the development of
Quantum
Mechanics which is today regarded as the
fundamental theory of Nature.
Some key events/observations that led to the
development of quantum mechanics…
_________________________________
Black body radiation spectrum (Planck,
1901)
• Photoelectric effect (Einstein, 1905)
• Model of the atom (Rutherford, 1911)
• Quantum Theory of Spectra (Bohr, 1913)
Some key events/observations that led to the
development of quantum mechanics…
_________________________________
• Matter Waves (de Broglie 1925)
Planck
Einstein
Bohr
Rutherford
de Broglie
Hydrogen Energy Levels
The basic hydrogen energy level structure is in agreement with the Bohr model. Common pictures
are those of a shell structure with each main shell associated with a value of the principal quantum
number n.
This Bohr model picture of the orbits has some usefulness for visualization so long as it is realized that the
"orbits" and the "orbit radius" just represent the most probable values of a considerable range of values.
The Bohr model for an electron transition in hydrogen between quantized energy levels with different
quantum numbers n yields a photon by emission, with quantum energy
1
me4  1
1 
1 
h 
 2   13.6  2  2  eV
2 2  2
8 0 h  n1 n2 
 n1 n2 
This is often expressed in terms of the inverse wavelength or "wave number" as follows:
me4
RH 
2
8 0 ch3
RH  1.097 107 m1
Uncertainty Principle
• We assume that if we measure something
we will have some small errors
• You know this from your lab work
• With better instruments and techniques
you can reduce these errors
• Heisenberg showed that there is a limit to
how small you can make the error!
Uncertainty Principle
Recall the diffraction limit of light.
We can measure position to about a
wavelength of the light we use. To
get more accurate position, shorten
the wavelength which ups the
frequency. But E=hf, so the photon
has higher energy. Whacks the
electron harder and you don’t know
where it goes or how fast it is
moving. Lower the frequency and
you get more uncertainty in position.
Uncertainty Principle
x  
p  h / 
xp  h
xp  h /2
Schrödinger Equation
The Schrödinger equation plays the role of Newton's laws and conservation of energy in classical
mechanics - i.e., it predicts the future behavior of a dynamic system. It is a wave equation in terms of
the wavefunction which predicts analytically and precisely the probability of events or outcome. The
detailed outcome is not strictly determined, but given a large number of events, the Schrödinger
equation will predict the distribution of results.
The kinetic and potential energies are transformed into the Hamiltonian which acts upon the
wavefunction to generate the evolution of the wavefunction in time and space. The Schrödinger
equation gives the quantized energies of the system and gives the form of the wavefunction so that
other properties may be calculated.
Time-independent Schrödinger Equation
For a generic potential energy U the 1-dimensional time-independent Schrodinger equation is
In three dimensions, it takes the form
for cartesian coordinates. This can be written in a more compact form by making use of the Laplacian
operator
The Schrodinger equation can then be written:
HΨ = EΨ
Time Dependent Schrödinger Equation
The time dependent Schrödinger equation for one spatial dimension is of the form
For a free particle where U(x) =0 the wavefunction solution can be put in the form of
a plane wave
k
2


2
p
h

2π
2
 2 
E
T
h
For other problems, the potential U(x) serves to set boundary conditions on the spatial part
of the wavefunction and it is helpful to separate the equation into the time-independent
Schrödinger equation and the relationship for time evolution of the wavefunction
Time-Independent Schrödinger
Equation
• Schrödinger developed the equation from
which we can find the wavefunction
• Below is time-independent Schrödinger
equation, which describes stationary states
– the energy of such states does not change
with time
• ψ(x) is often called eigenfunctions or
eigenstate
   x 

 U( x) x  E x
2
2m  x
2
2
Particle in a box with “Infinite Barriers”
• A particle is confined to a onedimensional region of space
between two impenetrable walls
separated by distance L
– This is a one- dimensional “box”
• The particle is bouncing
elastically back and forth between
the walls
– As long as the particle is inside
the box, the potential energy does
not depend on its location. We
can choose this energy value to be
Particle in a box with “Infinite Barriers”
• Since the walls are impenetrable,
there is zero probability of finding
the particle outside the box. Zero
probability means that ψ(x) = 0, for
x < 0 and x > L
• The wave function must also be 0 at
the walls (x = 0 and x = L), since the
wavefunction must be continuous
– Mathematically, ψ(0) = 0 and
ψ(L) = 0
Schrödinger Equation Applied to a
Particle in a “Infinite” Box
 2  2 x

 U( x) x  E x
2
2m  x
• In the region 0 < x < L,
where U(x) = 0, the
Schrödinger equation can
be expressed in the form
• We can re-write it as
 2  2 x 

 E x 
2
2m  x
 2 x 
2mE


 x 
2
2
 x

 2 x 
2


k
 x 
2
 x
2mE
k2  2

Schrödinger Equation Applied to a
Particle in a “Infinite” Box
 2 x
2
  k  x 
2
 x
• The most general solution to this differential
equation is
ψ(x) = A sin kx + B cos kx
– A and B are constants determined by the
properties of the wavefunction as well as
boundary and normalization conditions
Schrödinger Equation Applied to a
Particle in a “Infinite” Box
1.
Sin(x) and Cos(x) are finite and single-valued
functions
2. Continuity: ψ(0) = ψ(L) = 0
• ψ(0) = A sin(k0) + B cos(k0) = 0  B = 0  ψ(x) =
A sin(kx)
• ψ(L) = A sin(kL) = 0  sin(kL) = 0  kL = πn, n =
±1, ± 2, ± 3, …
kn 

L
n  k n 
  2 2
En  
2
2
mL

2mE n    2

  n
2

 L
2
2
 2  h2
n  

 8mL 2


 2
n


Schrödinger Equation Applied to a
Particle in a “Infinite” Box
• The allowed wave functions are given by
 n 
ψn(x)  A sin
x
 L 
– After, the normalization, the normalized
wave function
ψn(x) 
2
 n
sin
L
 L

x

Particle in the Well with Infinite Barriers
En 
 2 2
n 2  E0 n 2 , with E0 
 2 2
2mL 2
2mL 2
ground state (n  1) energy , E1  E0
Probability to Find particle
Finite Potential Well
Graphical Results for ψ (x)
• Outside the
potential well,
classical physics
forbids the presence
of the particle
• Quantum mechanics
shows the wave
function decays
exponentially to
Finite Potential Well
Graphical Results for Probability
Density, | ψ (x) |2
• The probability
densities for the
lowest three states
are shown
• The functions are
smooth at the
boundaries
• Outside the box, the
probability to find
the particle