The Schrodinger Equation Reading:OGC:(4.6)and (5.1)
The Schrödinger Equation Reading: OGC: (4.6) and (5.1) III-1
The Person Behind The Science Erwin Schrodinger 1887-1961 Highlights Born and educated in Vienna Received Nobel Prize in Physics with Paul Dirac (1933) Moments in a Life In 1927 Schrodinger moved to University of Berlin as Planck's successor Develops his wave equation in 1926 -2
The Person Behind The Science Erwin Schrödinger 1887-1961 In 1927 Schrödinger moved to University of Berlin as Planck's successor Develops his wave equation in 1926 Moments in a Life Highlights Born and educated in Vienna Received Nobel Prize in Physics with Paul Dirac (1933) III-2
The Schrodinger Equation HΨ=EΨ .H is the Hamiltonian Operator;you can't "cancel"the y 一“Cancelling”the平is like“cancelling”the x in f(x)=mx. You just can't do it. Our goal is to operate on (using the H Operator)and get an energy (E)multiplied by the same y. -3
The Schrödinger Equation HΨ = EΨ H is the Hamiltonian Operator; you can’t “cancel” the Ψ — “Cancelling” the Ψ is like “cancelling” the x in f(x) = mx. You just can’t do it. Our goal is to operate on Ψ (using the H Operator) and get an energy (E) multiplied by the same Ψ. III-3
Deriving the Schrodinger Equation .This equation describes the energy of an electron: Total Energy Kinetic Energy Potential Energy E=KE+PE Start with this classical equation. Use classical and quantum mechanical relationships to find the Hamiltonian Operator(H). Find values of that fit the Schrodinger Equation: HΨ=EΨ. -4
Deriving the Schrödinger Equation Total Energy = Kinetic Energy + Potential Energy E = KE + PE This equation describes the energy of an electron: Start with this classical equation. Use classical and quantum mechanical relationships to find the Hamiltonian Operator (H). Find values of Ψ that fit the Schrödinger Equation: HΨ = EΨ. III-4
Describing Kinetic Energy(KE) Classically: Quantum Mechanically: -ihaΨ KE-1mv2 Px= 2πdx 2 Iffx,y,z）=x2+y3+z4, p=mv,so KE= p2 2m then a证=2xand 0 ay 3y2 Combining Equations: KEx p_82y-h2 (close enough for now!) 2m 0x2 8mn2 ,2如 KE-KEx+KEy+KEz-8m2 x2+y2 0z2 g产2 )(V is a form of mathematical shorthand notation -5
∂f ∂ x ∂ f ∂ y — If f(x,y,z) = x2+y3+z4 , Describing Kinetic Energy (KE) Classically: Quantum Mechanically: then = 2x and = 3y2 px = −ih 2π ∂Ψ KE = ∂x 1 2 mv 2 p = mv , so KE = p 2 2 m Combining Equations: (close enough for now!) = - h 2 8 m π 2 ∇ 2 ( Ψ ) ( ∇ is a form of mathematical shorthand notation ) KE x = p x 2 2 m = ∂ 2 Ψ ∂ x 2 -h 2 8 m π 2 So KE =KE x + KE y +KE z = -h 2 8 m π 2 ∂ 2 Ψ ∂ x 2 + ∂ 2 Ψ ∂ y 2 + ∂ 2 Ψ ∂ z 2 III-5