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# lecture16 - 5.61 Fall 2007 Separable Systems page 1 QUANTUM...

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Unformatted text preview: 5.61 Fall 2007 Separable Systems page 1 QUANTUM IN : SEPARABLE SYSTEMS 1D Systems 3D Systems x ˆ r ˆ = ( x ˆ y ˆ z ˆ ) = i x ˆ + j y ˆ + k z ˆ ¡ d ¡ ∂ ¡ ∂ ¡ ∂ + j + k p ˆ = i dx p ˆ = ( p ˆ x p ˆ y p ˆ z ) = i i ∂ x i ∂ y i ∂ y ⎡ x ˆ, p ˆ ⎤ i ¡ ⎡ ⎤ ⎡ ⎤ ⎣ ⎦ = ⎡ ⎣ x ˆ, p ˆ x ⎤ ⎦ = i ¡ ⎣ y p ˆ, ˆ y ⎦ = i ¡ ⎣ z ˆ, p ˆ z ⎦ = i ¡ ˆ p ˆ 2 − ¡ 2 d 2 ˆ p ˆ 2 − ¡ 2 ∂ 2 − ¡ 2 ∂ 2 − ¡ 2 ∂ 2 T = = T = = + + 2 m 2 m dx 2 2 m 2 m ∂ x 2 2 m ∂ y 2 2 m ∂ y 2 ψ ( x ) ψ ( x y z , , ) O ˆ = ∫ ψ * ( x ) O ˆ ψ ( x ) dx O ˆ = ∫ ψ * ( x y z , , ) O ˆ ψ ( x y z , , ) dx dy dz By fiat, operators corresponding to different axes commute with one another. ˆˆ = ˆˆ p y ˆ = yp p p ˆ = ˆ ˆ etc . xy yx ˆ ˆˆ ˆ p p z z z x x z Further, operators in one variable have no effect on functions of another: xf ˆ ( y ) = f ( y ) ˆ p f ( x ) = f ( x ) ˆ f * ( z ) p = ˆ * ( ) x ˆ z p z ˆ x p x f z etc . The Time Independent Schrödinger Equation becomes: ⎡ ¡ 2 ⎛ ∂ 2 ∂ 2 ∂ 2 ⎞ ⎤ ⎢ − ⎜ 2 + 2 + 2 ⎟ + V ( x ˆ , y ˆ , z ˆ ) ⎥ ψ ( x , y , z ) = E ψ ( x , y , z ) ⎣ 2 m ⎝ ∂ x ∂ y ∂ z ⎠ ⎦ ∇ 2 the Laplacian ⎡ ¡ 2 2 ⎤ ⇒ ⎢ − ∇ + V ( x ˆ , y ˆ , z ˆ ) ⎥ ψ ( x , y , z ) = E ψ ( x , y , z ) ⎣ 2 m ⎦ H ˆ = − ¡ 2 ∇ 2 + V ( x ˆ, ˆ, ˆ ) y z Hamiltonian operator in 3D 2 m ˆ ( , y z , ) = ψ ( x , , ) 3D Schrödinger equation H ψ x E y z (Time Independent) Separation of variables 5.61 Fall 2007 Separable Systems page 2 IF V ( x ˆ, y ˆ, z ˆ ) = V x ( x ˆ ) + V y ( y ˆ ) + V z ( z ˆ ) ⎡ 2 ∂ 2 ⎤ ⎡ 2 ∂ 2 ⎤ ⎡ 2 ∂ 2 ⎤ ( ) ⎢ 2 x ( ) ⎥ ⎢ 2 y ( ) ⎥ ⎢ 2 z ( ) ⎥ ˆ , − x y + − + V ˆ H x , y z = + V ˆ + − + V ˆ z then ⎣ 2 m ∂ x ⎦ ⎣ 2 m ∂ y ⎦ ⎣ 2 m ∂ z ⎦ = H ˆ + H ˆ + H ˆ x y z ⇒ Schrödinger’s Eq. becomes: ⎡ H ˆ + H ˆ + H ˆ ⎤ ψ x , y , z = E ψ x , , ⎣ x y z ⎦ ( ) ( y z ) Then try solution of form ψ ( x , y , z ) = ψ x ( x ) ψ y ( y ) ψ z ( z ) (separation of variables) Where we assume that the 1D functions satisfy the appropriate 1D TISE: H ˆ ψ ( x ) = E ψ ( x ) x x x x H ˆ y ψ y ( y ) = E y ψ y ( y ) H ˆ ψ ( z ) = E ψ ( z ) z z z z First term: H ˆ x ψ x ( x ) ψ y ( y ) ψ z ( z ) = ψ y ( y ) ψ z ( z ) H ˆ x ψ x ( x ) = ψ y ( y ) ψ z ( z ) E x ψ x ( x ) E ψ ( ) ( ) ψ z = x x x ψ y y z ( ) Same for H ˆ and H ˆ ⇒ y z H ˆ ψ = E ψ ⎣ H x + H y + H z ⎦ ⎣ ψ x ( x ) ψ y ( y ) ψ z ( z ) ⎦ = ( E x + E y + E z ) ⎣ ψ x ( x ) ψ y ( y ) ψ z ( z ) ⎦ ⎡ ˆ ˆ ˆ ⎤ ⎡ ⎤ ⎡ ⎤ E = E + E + E x y z Thus, if the Hamiltonian has this special form, the eigenfunctions of the 3D Hamiltonian are just products of the eigenfunctions of the 1D Hamiltonian and the situation is equivalent to doing three separate 1D problems....
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## This note was uploaded on 09/24/2011 for the course MATH 1090 taught by Professor Greenwood during the Spring '08 term at MIT.

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lecture16 - 5.61 Fall 2007 Separable Systems page 1 QUANTUM...

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