Lecture12_QM4 - Quantum Theory Quantum 4 1 Topics q The 1-D...

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1 Quantum Theory Quantum Theory 4 4
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2 Topics Topics The 1-D Square Well The 1-D Harmonic Oscillator Quantum Tunneling Summary
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3 Recap Recap 1887 – 1961 Ψ = Ψ H t it ) ( 2 2 2 r V m H + - = ħ = h/2p In late 1925, building on de Broglie’s idea, Erwin Schrödinger proposed the following wave equation is a complex number called the wavefunction Ψ 1887 – 1961
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4 Recap Recap p ( x ) = ψ 2 = ψ ( ξ 29 ψ ( ξ 29 p ( x ) dx is the probability to find a particle between x and x + dx x x x + dx p ( x ) Max Born 1882 – 1970
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5 The 1-D Square Well The 1-D Square Well
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6 The 1-D Square Well The 1-D Square Well Ψ + Ψ - = Ψ ) ( 2 2 2 2 x V x m t i t t = ) ( x V = ) ( x V 0 ) ( = x V An object of mass m in a 1-D well with impenetrable walls
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7 The 1-D Square Well The 1-D Square Well Ψ + Ψ - = Ψ ) ( 2 2 2 2 x V x m t i t t If the potential energy function V ( x ) is independent of time, we can write the solution of the Schrödinger equation in the separable form ( , ) ( ) ( ) x t x t ψ ϕ Ψ =
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8 The 1-D Square Well The 1-D Square Well - η 2 2 μ δ 2 ψ ( ξ 29 δξ 2 + ς ( ξ 29 ψ ( ξ 29 = Ε ψ ( ξ
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