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Quantum mechanics review. Reading for week of 1/28-2/1 Chapters 1, 2, and 3.1,3.2 Reading for week of 2/4-2/8 Chapter 4. Schrodinger Equation (Time-independent). where.

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Schrodinger Equation (Time-independent)

where

The solutions incorporate boundary conditions and are a family of eigenvalues with increasing energy and corresponding

eigenvectors with an increasing number of nodes.

The solutions are orthonormal.


Physical properties: Expectation values

Dirac notation or bra-ket notation


Physical properties: Hermitian Operators

Real Physical Properties are Associated with Hermitian Operators

Hermitian operators obey the following:

The value <A>mn is also known as a matrix element, associated with solving the problem of the expectation value for A as the eigenvalues of a matrix indexed by m and n


Zero order models:

Particle-in-a-box: atoms, bonds, conjugated alkenes, nano-particles

Harmonic oscillator: vibrations of atoms

Rigid-Rotor: molecular rotation; internal rotation of methyl groups, motion within van der waals molecules

Hydrogen atom: electronic structure

Hydrogenic Radial Wavefunctions


Particle-in-a-3d-Box

V(x) =0; 0<x<a

V(x) =∞; x>a; x <0

b y ; c  z

V(x)

x

a

nx,y,z = 1,2,3, ...


Particle-in-a-3d-Box

V(x) =0; 0<x<a

V(x) =∞; x>a; x <0

b y ; c  z

V(x)

x

a


Zero point energy/Uncertainty Principle

In this case since V=0 inside the box E = K.E.

If E = 0 the p = 0 , which would violate the uncertainty principle.


Zero point energy/Uncertainty Principle

More generally

Variance or rms:

If the system is an eigenfunction of then is precisely determined and there is no variance.


Zero point energy/Uncertainty Principle

If the commutator is non-zero then the two properties cannot be precisely defined simultaneously. If it is zero they can be.


Harmonic Oscillator 1-d

F=-k(x-x0)

Internal coordinates; Set x0=0


Harmonic Oscillator Wavefunctions

Hermite polynomials

V = quantum number = 0,1,2,3

Hv = Hermite polynomials

Nv = Normalization Constant


http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc5.html#c1http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc5.html#c1


Raising and lowering operators:http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc5.html#c1

Recursion relations used to define new members in a family of solutions to D.E.


Rotation: Rigid Rotorhttp://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc5.html#c1


Rotation: Rigid Rotorhttp://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc5.html#c1

Wavefunctions are the spherical harmonics

Operators L2 ansd Lz


Degeneracyhttp://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc5.html#c1


Angular Momemtum operators the spherical harmonicshttp://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc5.html#c1

Operators L2 ansd Lz


Rotation: Rigid Rotorhttp://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc5.html#c1

Eigenvalues are thus:

l = 0,1,2,3,…


Lots of quantum mechanical and spectroscopic problems have solutions that can be usefully expressed as sums of spherical harmonics.

e.g. coupling of two or more angular momentum

plane waves

reciprocal distance between two points in space

Also many operators can be expressed as spherical harmonics:

The properties of the matrix element above are well known and are zero unless

-m’+M+m = 0

l’+L+l is even

Can define raising and lowering operators for these wavefunctions too.


The hydrogen atom solutions that can be usefully expressed as sums of spherical harmonics.

Set up problem in spherical polar coordinates. Hamiltonian is separable into radial and angular components


n solutions that can be usefully expressed as sums of spherical harmonics.

the principal quantum number, determines energy

l

the orbital angular momentum quantum number

l= n-1, n-2, …,0

m

the magnetic quantum number

-l, -l+1, …, +l


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