{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Chapter4_all

# Chapter4_all - PHY4604 R D Field Quantum Mechanics in Three...

This preview shows pages 1–4. Sign up to view the full content.

PHY4604 R. D. Field Department of Physics Chapter4_1.doc University of Florida Quantum Mechanics in Three Dimensions Schrödinger Equation: In three dimensional space we have dt t r d i t r H op ) , ( ) , ( r h r Ψ = Ψ where ) ( ) ) ( ) ( ) (( 2 1 ) ( 2 1 2 2 2 2 r V p p p m r V p m H op z op y op x op op r r + + + = + = and x i p op x = h ) ( y i p op y = h ) ( z i p op z = h ) ( Thus Schrödinger’s equation becomes dt t r d i t r r V t r m op ) , ( ) , ( ) ( ) , ( 2 2 2 r h r r r h Ψ = Ψ + Ψ where 2 op is the Laplacian operator 2 2 2 2 2 2 2 z y x op + + = and 2 2 2 op op p = h . Stationary State Solutions: The stationary state solutions are of the form h r r / ) ( ) , ( iEt e r t r = Ψ ψ and the time independent equation is ) ( ) ( ) ( ) ( 2 2 2 r E r r V r m op r r r r h ψ ψ ψ = + . Canonical Commutation Relations: It is easy to see that 0 ] ) ( , ) [( 0 ] ) ( , ) [( ] ) ( , ) [( = = = op j op i op j op i ij op j op i x x p p i x p δ h where δ ij is the Kroenecker delta function (i = 1,2,3 j = 1,2,3 δ ij = 0 if i j and δ ij = 1 if i = j, p 1 = p x , p 2 = p y , p 3 = p z , x 1 = x, x 2 = y, x 3 = z) and i op i x i p = h ) ( . Probability Density: The probability density is 2 | ) , ( | ) , ( t r t r r v Ψ = ρ and the probability of finding the particle between r r and r d r r r + at time t is r d t r 3 ) , ( v ρ with 1 ) , ( 3 = allspace r d t r v ρ .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
PHY6404 R. D. Field Department of Physics Chapter4_2.doc University of Florida The Angular Momentum Operator (1) The Momentum Operator in 3 Dimensions: In “position space” with Cartesian coordinates we have z i p y i p x i p op z op y op x = = = h h h ) ( ) ( ) ( and 2 2 2 2 2 2 ) ( ) ( ) ( ) ( op op z op y op x op p p p p = + + = h where 2 2 2 2 2 2 2 z y x op + + = . Angular Momentum: Angular momentum is the vector operator given by op z op y op x op op p p p z y x z y x p r L ) ( ) ( ) ( ˆ ˆ ˆ = × = r r r Hence, op y op z op x p z p y L ) ( ) ( ) ( = op z op x op y p x p z L ) ( ) ( ) ( = op x op y op z p y p x L ) ( ) ( ) ( = and in “position space” with Cartesian coordinates we have = y z z y i L op x h ) ( = z x x z i L op y h ) ( = x y y x i L op z h ) ( Commutation Relations: The commutator of, for example, L x and L y is ( ) op z op x op y op y op z op x op z op y op z op z op y op y op x op x op y op z op z op z op z op z op x op x op z op z op y op x op y op z op z op x op z op z op x op y op z op y op x L i p y p x i p p z x p z p y p p x z p x p z p p z z p z p z p p x y p x p y p p z y p z p y p x p z p z p z p x p y p z p y p x p z p z p y L L ) ( ) ( ) ( ) ]( ) ( , [ ) ]( , ) [( ) ]( ) ( , [ ] ) ( , ) [( ) ]( ) ( , [ ] ) ( , ) [( ) ]( ) ( , [ ] ) ( , ) [( ) ]( ) ( , [ ] ) ( , ) [( ] ) ( , ) ( [ ] ) ( , ) ( [ ] ) ( , ) ( [ ] ) ( , ) ( [ ] ) ( ) ( , ) ( ) ( [ ] ) ( , ) [( h h = = + = + + + = + = = We see that the commutator of any two of the angular momentum operators gives the third angular momentum operator as follows: op z op y op x L i L L ) ( ] ) ( , ) [( h = op x op z op y L i L L ) ( ] ) ( , ) [( h = op y op x op z L i L L ) ( ] ) ( , ) [( h = This can be summarized by the following: op k ijk op j op i L i L L ) ( ] ) ( , ) [( ε h = Note: ε iik = ε ijj = ε iji = 0, ε 123 = ε 231 = ε 312 = 1, ε 213 = ε 132 = ε 321 = -1.
PHY4604 R. D. Field Department of Physics Chapter4_3.doc University of Florida

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 29

Chapter4_all - PHY4604 R D Field Quantum Mechanics in Three...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online