Download Quantum Mechanics Lecture Notes: Harmonic Oscillator and Particle in Finite Box and more Summaries Advanced Physics in PDF only on Docsity! Modern Physics Lecture 17
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JMY Previous Lecture Schrodinger equation Particle in a finite box (finite potential barrier) Tunneling! Discussed in detail in FilesLecture materials Document: chap05B_schrodinger_v4.pdf Document: chap05C_SE_particle_box_v3.pdf Document: chap05D_SE_harmonic_v3.pdf PHY3101: Chapter 5 JMY Classical Harmonic Oscillator (“Spring”) Classical harmonic oscillator: Remember that Work Done is the integral of Force x distance PHY3101: Chapter 5 Potential Energy: Energy conservation means that: E = (constant) JMY Classical Harmonic Oscillator PHY3101: Chapter 5 E E determines range of motion! JMY Classical Harmonic Oscillator Why do we care about HO potential? It’s everywhere! Any potential with a minimum looks like a HO near the minimum PHY3101: Chapter 5 (radial potential) JMY QM HO: Behavior of Solutions Let’s study the behavior of SE for large x (next slide) Exponential fall: bends away from x axis outside classical xmax PHY3101: Chapter 5 JMY QM HO: Determine Form of Solution What happens when x is extremely large? Equation approaches This implies that solution falls rapidly like So let’s try solution of form This is a common approach when exponentials presentPHY3101: Chapter 5 JMY QM Wavefunctions for HO This solution works! Solutions are (for energy level En) Hn(x) are “Hermite polynomials”, En are evenly spaced! PHY3101: Chapter 5 2 n = 0, 1, 2.. 0 0 0 0 JMY HO Wavefunctions: n = 0 – 4 PHY3101: Chapter 5 Scaled x JMY HO Probability Distributions : n = 0 – 4 PHY3101: Chapter 5 Scaled x JMY Harmonic Oscillator Probability Calculation PHY3101: Chapter 5 First, look at the general form for the NORMALIZED wavefunction **NOTE: previous slides say “H( This means the Hermitian of ( (sqrt is only for ) 0 JMY An aside - Comparison to Classical HO Find fraction of total period particle is within [x, x + dx] Compare with classical occupation probability Agreement is poor for small n but improves steadily as n increases “Correspondence” principle: QM → classical as n → ∞ PHY3101: Chapter 5 =0
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JMY Particle In Infinite 3-D Box 3-D Schrodinger equation Postulate wavefunction solution using separation of variables PHY3101: Chapter 5 This what we previously had in 1D, and U(x)=0 in the box JMY Particle In Infinite 3-D Box continued Apply same arguments in 1D case for each coordinate Each energy term must be positive: Boundary conditions give , etc. Full solution is product of sin(kxx), sin(kyy), sin(kzz) (Next slide) ⟹ ⟹ PHY3101: Chapter 5 JMY Particle In Infinite 3-D Box continued States are labeled by the 3 integers nx, ny, nz ≥ 1 , , etc. with energies E111, E211, etc Multiple states can have same energy “degeneracy”⟹ In 1 slide: Degeneracies for 3D box In 2 slides: 2D box plots for , , PHY3101: Chapter 5
