Quantum Harmonic Oscillator:Analytic method
- 1. Quantum Harmonic Oscillator PRAMOD A C 1st M.Sc DoS in Physics University of Mysore Manasagangotri
- 2. Contents • Introduction • Classical harmonic oscillator • Quantum harmonic oscillator • Comparison • Applications • Conclusion • Reference 1
- 3. Introduction • The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. • Quantum harmonic oscillator provides a useful model for a variety of vibrational phenomena that are encountered for instance in classical mechanics, electrodynamics, statistical mechanics, solid state, atomic, nuclear, and particle physics. 2
- 4. Classical Harmonic Oscillator Figure 1: Classical Harmonic Oscillator Oscillator source: https://www.shutterstock.com This system exhibits sinusoidal behavior. From Hook’s law, the mathematical expression for the restoring force is, F = −kx (1) 3
- 5. Totle Energy of the system Figure 2: Classical hormonic oscillator source: https://images.app.goo.gl/pN7ibGogwmQbKCwM7 The kinetic energy, T = 1 2 mv2 0 cos2 r k m t ! (2) 4
- 6. Total Energy of the system The potential energy of the system is, V = 1 2 mv2 0 sin2 r k m t ! (3) then the total energy of the system is, E = 1 2 mv2 0 (4) The total energy of the classical harmonic oscillator is a constant value, which depends on v0 that can be any number. But it is not the case for quantum harmonic oscillator. 5
- 7. Quantum harmonic oscillator Figure 3: Quantum Harmonic Oscillater source : https : //www.google.com/imgres?imgurl = https 6
- 8. Quantum Harmonic Oscillator The Hamiltonian of a particle is, Ĥ = 1 2m p̂2 + (mwx)2 (5) To find the energy eigenvalues and eigenstates of this Hamiltonian we can use two methods, • Analytic method • Ladder or Algebraic method Then slove the time independent Schrodinger equation, − ℏ2 2m d2 ψ dx2 + 1 2 mω2 x2 ψ = Eψ (6) 7
- 9. Quantum Harmonic Oscillator By analytic method get the wave function, Ψn(x) = 1 p√ π2nn!x0 e−x2/2x2 0 Hn x x0 (7) where Hn are nth order polynomials called Hermite polynomials. Hn(y) = (−1)n ey2 dn dyn e−y2 (8) The energy levels of the quantum harmonic oscillator are, E = n + 1 2 ℏω (9) where (n = 0, 1, 2, 3....) The above equation represents lowest allowed energy range. 8
- 10. Energy states of different particle or molecules Figure 4: Energy states of different particle or molecules 9
- 11. Probability distribution Figure 5: Probability distributionsource : http://hyperphysics The solution of the Schrodinger equation for the quantum harmonic oscillator gives the probability distributions for the quantum states of the oscillator. 10
- 12. Probability distribution comparison Figure 6: Probability distribution comparisonsource : http://hyperphysics 11
- 13. Probability distribution comparison Figure 7: Probability distribution comparison source : http://hyperphysics 12
- 14. Applications • Molecular vibrations • Lattice vibration • Thermal vibration in superconductors 13
- 15. Molecular vibrations Figure 8: energy levels source : http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/imgqua/qhar.gif 14
- 16. Molecular vibration The energy levels are quantized at equally spaced values.the angular frequency is given by, ω = s k µ where µ = m1m2 m1 + m2 is the reduced mass and m1 and m2 are the masses of the two atoms and k is bond force constant. 15
- 17. Molecular vibration • This form of the frequency is the same as that for the classical simple harmonic oscillator. • The transition energy is nearly equal to ℏω • The most surprising difference for the quantum case is called ”zero-point vibration” of the n=0 ground state. This implies that molecules are not completely at rest, even at absolute zero temperature. 16
- 18. Conclusion • The quantum harmonic oscillator is one of the foundation problems of quantum mechanics. • Is the foundation for the understanding of complex modes of vibration in larger molecules, the motion of atoms in a solid lattice etc. 17
- 19. Reference Books • Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 978-0-13-805326-0. • Zettili, N. (2003). Quantum mechanics: concepts and applications. Online sources • https://blog.cupcakephysics.com/thermodynamics • ”Quantum Harmonic Oscillator”. Hyperphysics. Retrieved 24 September 2009 • https://blog.cupcakephysics.com/thermodynamics 18