Schroedinger Equation III

Probability Current Density: In general for 3 space dimensions the probability density
P(r, t)=ψ*ψ dτ where dτ is a volume element. Integrating P over all space gives us the total probability N ( if not normalized to 1).

Differentiating the integrated probability density w.r.t the time t and replacing the time derivatives of ψ* and ψ from the Schroedinger equation and its complex conjugate we have

∂/∂ t ∫_V P dτ= -i /2m ∫_V [ψ∇² ψ* - ψ*∇²ψ]

using the Green theorem on the RHS we can write the integrand as divergence of a vector field ∇.S where S= -/m Im (ψ∇&psi*). Thus we have the equation

∂/∂ t P = ∇.S which is a equation of continuity for the
probability and indicates that probability is conserved. Correspondingly S is called the Probability Current Density ( in analogy with the current density in EM theory or the velocity current in fluid mechanics). So we see that Probability
can flow. In particular for free particle we see that S=P v where
v is the velocity of the particle. This is just like J=ρ v in EM theory.

As an exercise do this for 1 dimensions Ref Gasiorowicz ( in 1 dim there is no Green Theorem and one will have to use total derivatives to reduce the integral).

Schroedinger Equation II

The postulates of Quantum Mechanics:

I. Dynamical variables or observables ( measurable quantities ) are associated with actions of Linear Operators. ( Note that some variables like spin are intrinisically quantum mechanical without any classical analogs).

** We will adopt the covention of representing operators by boldface

Assuming ψ(x,t)= A e^i(kx-ωt) for a free particle it is easy to see
by direct operation that the momentum operator p=-i ℏ(∂/∂ x)
and that for the energy operator E=-i ℏ(∂/∂ t)

II. The Measurement of a dynamical variable or observable A that yields the value a leaves the system in a quantum state given by the wave function Φ_a
or A Φ_a=a Φ_a is an eigenvalue equation.
For example pψ=pψ.

III) Expectation Value: The average value of an observable for a system in the quantum state given by the wave function ψ is given as
⟨ C ⟩=∫ψ*C&psi dx

The average value is understood in the sense of an ensemble average where
simultaneous measurements of the observable is made at time t on a large number of exact replica of the system with identical initial quantum state specified by ψ(x,0).

IV) The time development of the wave function ψ(x,t) is given by the time dependent Schroedinger's equation.

Schroedinger Equation and the Probability Wave Function I

http://www.youtube.com/watch?v=6Q4_nl0ICao

The above youtube link shows the thought experiment described in the class and also available in the first chapter of Feynman Lectures III. It clearly tells us that the waves describing matter are actually probability waves such that the modulus squared of this probability wave amplitude describes the probability of the particles location and behaves just like the intensity of a normal ( linear) wave ( like an EM Wave).

This may be summarized as

For normal waves Intensity I ~ |amplitude|²

For probability waves Probability P ~ |probability wave amplitude|²

So we may describe the matter waves by a Probability Wave function/Wave Function
( considering only 1 space dimension x for simplicity)
Ψ(x,t)=A e^i(kx-ωt) where A is a constant. This wave function
is expected to satisfy a linear wave equation so that the superposition principle holds and must be consistent with the following relations.

p=ℏk; , E=ℏω=P²/2m + V(x)
or E=ℏ ²K²/2m + V(x)=ℏω

From these relations and the form of the wave function Ψ(x,t) it is obvious that
the equation must be second order in space derivatives ( for the k² and
first order in the time derivative ( for ω). So we can assume a form of the equation to be

[ α∂²/∂ x² + V(x)]ψ (x,t)=β(∂/∂ t) ψ(x,t)

Using the form of ψ(x,t) this gives -αk² +V =∓iβω this has two solutions β=± i ℏ. Assuming the + sign ( - is equivalent) we have the Schroedinger equation in 1 space dimension as

[ -ℏ²/2m ∂²/∂ x² + V(x)]ψ (x,t)=i ℏ(∂/∂ t) ψ(x,t)

This can easily be generalised to 3 dimensions as

[ -ℏ²/2m ∇² + V(x)]ψ (r ,t)=i ℏ(∂/∂ t) ψ(r ,t)

This is refered to as the Time Dependent Schoredinger Equation

It can be shown that the solution to this equation ψ is a complex valued function of space and time. Hence ψ is NOT MESURABLE.
The Probability density of location of the particle between x to x + dx is then proportional to P(x,t)~ ψ². P(x,T) must be real and positive semi definite that is either positive or zero.

The state of the system is given by the wave function ψ and knowing the state
at time t=0 the Schroedinger equation predicts the state at time t=t₀. However notice that since all Physics is contained in |ψ|² the knowledge of all physical characteristic of the system at t=0 does not determine the function ψ(x, 0).

Quoting Max Born "Motion of the particles in QM conforms to the laws of probability however probability itself is propagated in CAUSAL FASHION through the Schroedinger equation.

The total probability is given as the volume integral over all space ∫|psi;|² d x and it must be finite so that the integral must converge. This requires the wave function to be a SQUARE INTEGRABLE or an L² function. The total probability may be normalized to 1.

Notice now that the probabilities corresponding to two wave functions are individually P₁ and P₂ but when they are superposed their
probability
P₁₂=|psi;₁ +psi;₂|²=P₁ + P₂ + Cross Terms. The cross terms gives rise to the interference pattern just like the cross terms in the intensity of superposed EM waves like light in a Youngs Double slit experiment described at the beginning.

Bohr Atom Model

http://hyperphysics.phy-astr.gsu.edu/hbase/bohr.html#c4

http://hyperphysics.phy-astr.gsu.edu/hbase/hyde.html

Click the relevant blobs for the webpages related to the index in the link below

http://hyperphysics.phy-astr.gsu.edu/hbase/bohrcn.html

TO BE UPDATED

Wave Groups and the Uncertainty Principle

http://en.wikipedia.org/wiki/Group_velocity

 http://www.youtube.com/watch?v=KT7xJ0tjB4A

http://hyperphysics.phy-astr.gsu.edu/hbase/uncer.html

TO BE UPDATED

Matter Waves

The host of experimental data and the theoretical ideas of Planck and Einstein conclusively established that radiation has a DUAL behavior. Namely it behaves both as
a wave in interference and diffraction and as a particle in photoelectric and compton effect. Inspired by this and guided by the close correspondence of the Fermats principle of optical path in geometrical optics ( particle like behavior) and the
Least Action principle in Classical Analytical Mechanics, de Broglie postulated that
Matter must also exhibit wave like behaviour.

Since for radiation E=hν and p=h/λ de Broglie postulated that the wavelength of matter waves was also given as λ_DB=h/p where p is the momentum of the particle. A quick calculation with standard values show that for macroscopic systems and velocities λ_DB is negligibly small, for a 1.0 kg mass moving at 10 m/s the λ_DB =6.6 x 10 ^-35  m which is too small to be detectable. But for a 100 eV energy electron it was of the order of 1 Angstrom.

From Optics we know that when λ >> a where a is the aperture dimension there is no difraction and geomterical optics holds. For diffraction λ ~ a.
So obviously to have observable diffraction effects λ_DB ~ a which is 1 angstorm. Such gratings are offered by the ordered periodic arrays of atomic layers in a crystal. The inter layer spacing is of the order of 1 angstorm.
So an electron beam scattered from such a crystal should exhibit diffraction.

This experiment was first performed by Davisson and Germer

the results of the experiment are summarized here

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/davger2.html#c1

The results clearly showed that the wave particle duality was valid also for matter just as in radiation and obeyed the same formula for both the momentum and energy namely E=hν and p=h/λ




Bohr's Complementarity Principle : This principle asserts that the wave and the particle behaviours for both matter and radiation are complementary. In a single experiment either the wave nature or the particle nature is manifest but never together.

So radiation and matter are neither simply waves nor simply particles. They are more general objects. In advanced applications the view of radiation and matter that emerges is a general description in terms of quantized fields which displays both particle (quanta) and wave ( field) properties. Particles are described as the field quanta


The link between the particle and the wave picture is provided by an interpretation of the wave particle duality based on probability. In the wave picture the Intensity I ~ E_av² where E_av is the average electric field ( amplitude) over 1 cycle. In the photon picture I ~ N_avhν where N_av is the average number of photos crossing unit area/unit time perpendicular to the propagation vector of the EM wave. In EM theory E_av² is proportional to the energy density. Einstein interpreted this as the average photon density which owing to the statistical nature of emission was related to the probability measure for a photon to cross unit area per unit time. So I~E_av² ~N_avhν .

Born borrowed this probability interpretation later to apply to the de Broglie's matter waves also. Since probability and intensity had a simmilar behaviour and the intensity is related to the square of an amplitude, one thinks of a probability wave
amplitude whose square is related to the probability. This wave is a probability wave
whose amplitude squared is related to the probability. This probability wave should satisfy a linear wave equation simmilar to the wave equation satisfied by a standard
EM wave for the superposition principle to be valid.