2303_-_Spr_2011_-_Week_12_-_QM_-_G

2303_-_Spr_2011_-_Week_12_-_QM_-_G - Week 12.1 Quantum...

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1 Week 12.1 Quantum Mechanics Homework Review Infinite Square Well Quantization of Energy Normalization Probability Density Taylor Ch. 7 EG
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2 with Infinite Square Well: Solve general Ψ from S.Eqn. Inside Well : (U = 0) Oscillator y Baski U(x)
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3 ψ cannot penetrate height barriers! Infinite Square Well: Solve general Ψ from S.Eqn. Outside Well : (U = 0 ) Baski U(x)
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4 U(x) Infinite Square Well: Boundary Conditions Boundary Conditions: Quantized Energy Solutions Baski Inside well:
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5 Infinite Square Well EG U U = 0
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6 so Infinite Square Well: Normalization Normalization Wave Solutions Baski
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7 n = 1 n = 3 ψ n(x ) ψ n2(x ) Infinite Square Well Wave Functions and Probability Energy Baski n = 2 U U = 0
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8 Quantum Mechanics For a 2D infinite square well, we get similar solutions. EG H.
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9 Week 12.2 Schrödinger Equation Free Electron Finite Quantum Well Homework 12 On Course Website Turn in problems on Monday in class EG Unlike conventional solar panels, solar cells based on nano-sized “quantum dots” can generate multiple electrons per photon, potentially increasing electrical output. – LANL (currently at 5% efficiency)
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10 Exponential Notation
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11 Full Time Dependent SWE The Wave function for a free particle is given by:
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12 Free Electron Baski For a free particle:
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13 Free Electron For a free electron, U(x) = 0, and so
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14 Free Electron For a free particle k is not quantized. Particle has any energy E ≥ 0. Wave function is imaginary (Can make standing waves by adding two opposing travelling waves.)
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15 Infinite Square Well An electron is in the n = 3 excited state of a 1-D infinite square well (width L). Draw the wave function and probability distribution of the electron. Solve the probability of finding the electron at x = 0.5 L in a width x = 0.02 L. Baski
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16 Finite Square Well Potential EG Look for Bound solutions, i.e. E < U0. Simple 1-D model for atoms. Energy Uo E
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17 where Finite Square Well Inside Well : (U = 0) Oscillator y Baski Energy Uo E
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18 (x) must be bounded, can penetrate barriers Then match boundary conditions (x) and > ´(x) must be continuous at x = 0 and x = L Finite Square Well where Outside Well : (U = Uo) Decaying Baski Energy Uo E
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19 n = 3 “leaks” outside barrier Ψ n(x ) Ψ n2(x ) Finite Square Well Potential: Visual Solutions Wave and Probability Solutions Energy Solutions E Uo E 3 E 2 E 1 Baski n = 2 n = 1
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20 Extras
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21 Week 12.3 Schrödinger Equation Finite Square Well Tunneling Simple Harmonic Oscillator EG
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