Chem 373- Lecture 2: The Born Interpretation of the Wavefunction

Lecture 2: The Born Interpretation of the Wavefunction This lecture covers the following parts of Atkins 11.4 The Born interpretation of the wavefunction Lectures on-line Born Interpretation (PDF) Born Interpretation (PowerPoint) Tutorials on-line The postulates of quantum mechanics (This is i s the writeup for Dry-lab-II)( This lecture coveres parts of postulate 2) Probability The wave function and probability The Development of Classical Mechanics Experimental Background for Quantum mecahnics Early Development of Quantum mechanics Audiovisuals on-line review of the Schrödinger equation and the Born postulate (PDF) review of the Schrödinger equation and the Born postulate (HTML) review of Schrödinger equation and Born postulate (PowerPoint **, 1MB) Slides from the text book (From the CD included in Atkins ,**)

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Quantum Mechanical Hamiltonian

From Last Lecture

ˆ is constructed The quantum quantum mechanical mechanical Hamiltonian Hamiltonian H constructed by the following following transforma transformation tionss :

HClass → x− y

>

ˆ = H

ˆ x

− > ˆy

; ;

1 2 2 2 ˆx + p ˆy + p ˆ z + V( ˆ p x, yˆ , zˆ ) 2 m

(

)

δ ˆx− > px − > p i δx h δ ˆy− > py − > p i δy h

Solve Schrödinger equation ˆ Ψ = EΨ H









Quantum Mechanical Hamiltonian Postulate 2 To every physical observable

Θclass

there corresponds a linear

Θ QM

Hermitian operator. In order

Θ class ( x, y, z, p x , p y,p z )

to obtain this operator,

ˆ Θ x , ˆy, ˆz, pˆ x , pˆ y,pˆ z ) QM ( ˆ

we write out the classica classicall - mechani mechanical cal expressi expression on for for the observable in terms of ˆ x

> x ;

p

> pˆ

>

h

δ

















Interpretation of th thee wav wavef efun unct ctio ion n in in 1 - D

Ψ = EΨ

ˆ H

Schrödinger Equation

1. The wavefunction contains all the dynamic information about the system it describes 2. The square modulus of the wavefunction at x is proporional to the probability of finding the particle at x If the wavefunction of a particle has the value Ψ( x ) at a point x, than the probability probability of finding the particle between x and x + ∆x is :









Interp Interpreta retatio tion n of the the wav wavefu efunct nction ion in 1 - D Ψ( x ) → pr prob obab abil ilit ity y ampl amplit itud udee positive positive,, negative negative,, complex complex | Ψ( x) |2 =

→

Ψ( x)Ψ(

x)*

always positive

c = a + ib a, b rea reall cc

*

*

(a + ib)(a + ib)











Interp Interpreta retatio tion n of the the wav wavefu efunct nction ion in 1 - D Ψ( x ) →

prob pr obab abil ilit ity y ampl amplit itud udee

positive, positive, negative nega complex 2 tive,, complex * | Ψ( x) | = Ψ( x)Ψ( x)











Interpretation of th thee wave wavefu func ncti tion on in 3 - D If the wavefunction of a particle has the value

Ψ( x, y, z) at a point (x,y,z) than the probability of finding the particle between x and x + ∆x; y and y + ∆y; z and z + ∆z is :





















Spher pheric ical al Coord oordin inat ates es We are going to make use of the spherical polar coordinate system Longitude Latitude Dis tan ce











Spher pheric ical al Coord oordin inat ates es We are going to make use of the spherical polar coordinate system Z (x,y,z) → (r, (r, Θ,φ

Θ

)













Spher pheric ical al Coord oordin inat ates es













Spher pheric ical al Coord oordin inat ates es













Normalization













Normalization













Normalization





















































































Chem 373- Lecture 2: The Born Interpretation of the Wavefunction

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