Wave function and Schrdinger equation
Fundamentals of quantum mechanical approach
Framework of classical mechanics of particles
Particles are indivisible and countable; can be presented as geometrical points A dynamics of a particle is described by a set
Lecture 17: Radial Wavefunction and Orbital Levels of the Hydrogen Atom The material in this lecture covers the following in Atkins. The structure and Spectra of Hydrogenic Atoms Section 13.1 The structure of hydrogenic atoms (b) The radial solution 13.2
Identical particles
Bosons and fermions
Two-particle systems
The system consisting of two particles is described by a wavefunction dependent on coordinates and spins of both particles: (r1 , r2 , t )
12
The Hamiltonian will include kinetic and potential e
Hydrogen atom
Classical mechanics in a Coulomb (gravitational potential)
Motion in the central potential in classical physics is characterized by conserving angular momentum
L = pr sin
Due to this conservation law momentum of the particle changes when it
Addition of Angular Momentum and spin
Total angular momentum
Lets consider states, in which a particle has both angular momentum and spin. Can we define something like total angular momentum? In order to define this operator, which can symbolically writte
Addition of spins
A two-particle system
Two spin particles
Consider a hydrogen atom in its ground state in a magnetic field. It consists of two particles electron and proton, each having spin , and interacting therefore with the magnetic field. What are p
Spin
Optical spectra in the presence of magnetic field
Splitting of energy levels due to Zeeman effect results in appearance of new lines in optical spectra. Indeed instead of transitions from a degenerate level, which are characterized by the same freque
Angular momentum
Algebraic theory
We consider several examples of the eigenfunctions. We begin with a wave function, which depends only on the distance from a center of the coordinate system.
Eigenfunctions of angular momentum: zero momentum
0,0 (r ) = R
Angular momentum
Basic Properties
Classical definition = r p L is converted into a QM operator by replacing coordinate and momentum by respective operators
Angular momentum in QM
Lx = ih y z ; Ly = ih z x ; y z x z L = rp L = ih x y x x y
All components
Axioms of quantum mechanics
Generalized Statistical Interpretation
Quantum states
Properties of quantum mechanical systems are described in terms of quantum states. A state can be described using different mathematical objects. Coordinate dependent wave f
Axioms of quantum mechanics
Generalized Statistical Interpretation
Quantum states
Properties of quantum mechanical systems are described in terms of quantum states. A state can be described using different mathematical objects. Coordinate dependent wave f
Harmonic oscillator
Algebraic approach
Classical harmonic oscillator
A block attached to an elastic spring is a typical example of a harmonic oscillator. One can find position as a function of time using Hooks law in combination with 2nd Newtons law
d 2x
Finite potential well
Continuous spectrum
Finite potential well: Continuous spectrum
V E
-a/2
a/2
I
II
V0
III
We again consider three different regions:
I x < a / 2 II a / 2 < x < a / 2 III x > a / 2
Scattering wave functions
In the case of E > 0 wave fun
Finite Quantum Well
Systems with two types of spectrum
Finite potential well: Two energy regions
V E>0
-a/2
x
a/2
V0
E<0
For E < 0 classical motion is bounded, in quantum mechanics states are normalizable and energy spectrum is discrete For E > 0 classica
Energyandtime
Hamiltonian
Classicalenergyinquantummechanicscamalsobeconvertedtoanoperatorcalled Hamiltonian 2 2
E=
p p + U ( x) H = + V ( x) 2m 2m
Inordertodefinetheexplicitformofthisoperatorweneedtodefinethesquareofan operator. 2
A f ( x) = AAf ( x) =
Queens College Department of Physics
Principles of Quantum Mechanics
Spring 20010 semester Prof. Lev Deych
Class time: T & Th. 4:30 6:20 PM Main textbook A.C. Phillips, Introduction to Quantum Mechanics, Wiley & Sons Ltd, 2003 Additional books D. Griffith