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Chapter7_all

# Chapter7_all - PHY3063 R D Field Quantum Mechanics in Three...

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PHY3063 R. D. Field Department of Physics Chapter7_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 ) ( op op i p = r h r with z z y y x x op ˆ ˆ ˆ + + = r . 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 op op + + = = r r 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 ij op j op i i x p δ h = ] ) ( , ) [( 0 ] ) ( , ) [( = op j op i p p 0 ] ) ( , ) [( = op j op i x x 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 , where dxdydx r d = 3 in Cartesian coordinates and φ θ θ d drd r r d sin 2 3 = in Spherical coordinates.

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PHY3063 R. D. Field Department of Physics Chapter7_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 =
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