One particle central force

One particle moving under a central force.

CENTRAL FORCE•Derive from potential energy function ,function only of distance of particle from origin.•Spherically symmetry. V=v(r)

The relation between force (F) and potential (P).

Since V is the function only on r

Hence

The equation become

QM of a single particle to a central force.

The Hamiltonian operator whereTransform these coordinate to

laplacian operator (square each operators).

This will result

Gives the operator for square of magnitude of orbital angular momentum of a single particle L^2.

The Hamiltonian becomes

In classical mechanics a particle subject to a central force has angular momentum conserve.

In quantum mechanics has definite values for both energy and angular momentum.

The commutator of H and L must be vanish.

Set the simultaneous eigenfunction of Ĥ, L^ and L^2 for a central-force problem.

Let the ψ denote of these common eigentfunction.

So, the Schrödinger equation become

Non interacting particles

The particles exert no force on each other and the classical-mechanical energy is sum of energy of two particles.

The Hamiltonian operator isWhere Ĥ1 only involve coordinate q1

and Ĥ2 only involve coordinate q2.The Schrödinger equation become

The solution of separation variables, we setting

When the system is composed of two no interacting particles, we can reduce two-particle problem into two separate one particle problem by

In general can solve any number of non interacting particles. For n such particles, we have

For system of non interacting particles, the energy is the sum of the individual energies of each particle and wave function is the product of wave functions for each particles.

This result also apply to single particle whose Hamiltonian is the sum of separate term for each coordinate

So we can conclude that the wave function and energies are

Reduction two particle to a one particle problem

Hydrogen has two particle, proton and electron with coordinate (x1,y1,z1) and (x2,y2,z2), the potential energy of interacting between these particles is a function of only the relative coordinate x2-x1, y2-y1,z2-z1 of particles.

In this case two particles problem can be simplified to two separate one particle problem.

We can draw vector R like this

The kinetic energy is the sum of kinetic energy of two particle

Let M is total mass and m is reduce mass

Thenthe 1st term is the kinetic energy due

to translational motion of the whole system of mass M. The quantity of kinetic energy of mass M at the centre of mass. note: translational is motion at the same displacement.

The 2nd term is kinetic energy of internal motion of two particles.(vibration and rotation)

The Hamiltonian function isThe quantum-mechanical energy isSo the translational energy EM found

by solving Schrödinger equationAnd the Eµ can be found using the

Schrödinger equation

Two particle rigid rotor

Two particle in the system that held at fixed distance from each other by a rigid massless rod.

The kinetic energy is wholly rotational energy.

So the energy of the rotor is wholly kinetic and V = 0.

The Hamiltonian equation become

Where m1 and m2 are the masses of two particles.

Hydrogen like atom

Discussing atom or molecule, considering isolate system

Ignoring interatomic and intermolecular interaction.

Treat just as hydrogen atom- 1 electron and a nucleus of charge Ze.

Hydrogen Z=1

Hydrogen like atom is the most important system in QM.

Schrödinger equation for atom with more than one electron can’t be obtained- interelectronic repulsion.

So we assume no repulsion so electron can be treated independently.

Atomic wave function will be approximate by a product of one electron function.

A one-electron wave function is called orbital.

An orbital for an electron in an atom is called atomic orbital.

Atomic orbital's is use to construct approximate wave function for many-electron atoms or for wave function for molecule.

Degeneracy

For bound state(transition state), the energy depends only on n. but wavefunction depend on all three quantum number n, l, and m.

• n = 1, 2, 3, …..• l = 0, 1, 2, 3, …,n – 1 • m = -l, -1, +1…0…l – 1 , l

Degeneracy - when different value of l or m but have same value of n and have the same energy. Except for n = 1.

For each value of l, the value of m is 2l+1.

The degree of degeneracy of bound state level of hydrogen like atom for discreet level is found n square.

For continuum levels for a given energy there is no restriction on the maximum value of l hence these levels are infinity-fold degenerate.(no discreet).

The bound energy of hydrogen like atom is discrete.

The change in n are allowed so the wave number of hydrogen spectral lines are