Electron wave function of first 3 states
- 1. • Review of quantum mechanical concept • Review of solid state physics Unit-I Prepared by S.Vijayakumar, AP/ECE Ramco Institute of Technology Academic year (2017-2018 odd sem)
- 2. • “I think that I can safely say that nobody understands quantum mechanics.”— Richard Feynman (Nobel Prize, 1965)
- 3. Erwin Rudolf Josef Alexander Schrödinger Born: 12 Aug 1887 in Erdberg, Vienna, Austria Died: 4 Jan 1961 in Vienna, Austria Nobel Prize in Physics 1933 "for the discovery of new productive forms of atomic theory"
- 4. An equation for matter waves? De Broglie postulated that every particles has an associated wave of wavelength: ph/=λ Wave nature of matter confirmed by electron diffraction studies etc (see earlier). If matter has wave-like properties then there must be a mathematical function that is the solution to a differential equation that describes electrons, atoms and molecules. The differential equation is called the Schrödinger equation and its solution is called the wavefunction, Ψ. What is the form of the Schrödinger equation ?
- 5. Electron wave function of first 3 states
- 6. Interpretation of Ψ(x,t) As mentioned previously the TDSE has solutions that are inherently complex ⇒Ψ (x,t) cannot be a physical wave (e.g. electromagnetic waves). Therefore how can Ψ (x,t) relate to real physical measurements on a system? The Born Interpretation dxtxPdxtxdxtxtx ),(),(),(),( 2* =Ψ=ΨΨ Ψ*Ψ is real as required for a probability distribution and is the probability per unit length (or volume in 3d). The Born interpretation therefore calls Ψ the probability amplitude, Ψ*Ψ (= P(x,t) ) the probability density and Ψ*Ψ dx the probability. Probability of finding a particle in a small length dx at position x and time t is equal to
- 7. Normalisation ∫∫ ∞ ∞− ∞ ∞− =Ψ= 1),()( 2 dxtxdxxP Total probability of finding a particle anywhere must be 1: This requirement is known as the Normalisation condition.
- 8. Energy band diagram of more than one electron of an atom Splitting of 3 energy states into allowed band of energy
- 9. Isolated silicon and silicon crytal Valence band electrons are losely bound N=1,2 are full occupied gets less interaction
- 10. Energy levels of interfacing atoms forms energy bands in solids
- 11. E-K relationship of the Kronig-Penny model and the energy band strcuture Electrons can take only alloed energy va
- 12. Potential function of single isolated atom and overlapping of adjacent atom
- 13. Net potential function of one dimensional single crytal One dimensional periodic function of kroneg penny mod
- 14. E-K diagram of reduced brilluion zone