Quantization of Lattice Waves: From Lattice Waves to Phonons, Lecture notes of Quantum Mechanics

Download Quantization of Lattice Waves: From Lattice Waves to Phonons and more Lecture notes Quantum Mechanics in PDF only on Docsity! 1 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Handout 20 Quantization of Lattice Waves: From Lattice Waves to Phonons In this lecture you will learn: • Simple harmonic oscillator in quantum mechanics • Classical and quantum descriptions of lattice wave modes • Phonons – what are they? ECE 407 – Spring 2009 – Farhan Rana – Cornell University Classical Simple Harmonic Oscillator    txm m tp E o x Total 22 2 2 1 2  x PE      txmxV m tp o x 22 2 2 1 ˆPE 2 KE  Consider a particle of mass m in a parabolic potential In quantum mechanics, the dynamical variables and observables become operators: The total energy is:     22 2 ˆ 2 1 2 ˆˆ ˆ ˆ xm m p HE ptp xtx o x Total xx    2 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Quantum Simple Harmonic Oscillator Review - I 22 2 ˆ 2 1 2 ˆˆ xm m p H o x  x PE   22 2 ˆ 2 1 ˆPE 2 ˆ KE xmxV m p o x  Consider a particle of mass m in a parabolic potential The quantum mechanical commutation relations are:   ipx x ˆ,ˆ Define two new operators: x o o p m ix m a ˆ 2 1 ˆ 2 ˆ     x o o p m ix m a ˆ 2 1 ˆ 2 ˆ     Hamiltonian operator is: ECE 407 – Spring 2009 – Farhan Rana – Cornell University Quantum Simple Harmonic Oscillator Review - II x o o p m ix m a ˆ 2 1 ˆ 2 ˆ     x o o p m ix m a ˆ 2 1 ˆ 2 ˆ     The quantum mechanical commutation relations are:     1ˆ,ˆˆ,ˆ  aaipx x  The Hamiltonian operator can be written as:         2 1 ˆˆˆ 2 1 2 ˆˆ 22 2 aaxm m p H oo x   The Hamiltonian operator has eigenstates that satisfy:n  .............3,2,1,0ˆˆ  nnnnaa nnnaanH oo               2 1 2 1 ˆˆˆ   5 ECE 407 – Spring 2009 – Farhan Rana – Cornell University From Classical to Quantum Description                    k j kjkj j j tRutRuRRK dt tRudM E ,, 2 1, 2 2   So we have finally:                  FBZ in *2 * ,, 2 ,, 2q tqUtqUq NM dt tqdU dt tqdUNM     Going from classical to quantum description:        nn nn Rp dt tRdu M RutRu    ˆ , ˆ,   The atomic displacements and the atomic momenta become operators: Commutation relations are:        iRpRu nn ˆ,ˆ Lattice wave amplitudes uncoupled in the PE term ECE 407 – Spring 2009 – Farhan Rana – Cornell University From Classical to Quantum Description The amplitudes of lattice waves are now also operators: The commutation relations for the lattice wave amplitudes are:           ','ˆ,ˆˆ,ˆ qqjj N i qPqUiRpRu      The Hamiltonian operator in terms of the lattice wave amplitude operators is:                 FBZ in 2 ,ˆ,ˆ 2 ˆ 2 ˆ q tqUtqUq NM qPqP M N H    can hold only if         tqUtqUetqUtRu q Rqi n n ,,,, * FBZ in .    Classical: Quantum:         tqUqUeqURu q Rqi n n ,ˆˆˆˆ FBZ in .             tqPtqPetqPtRp q Rqi n n ,,,, * FBZ in .    Classical: Quantum:         qPqPeqPRp q Rqi n n     ˆˆˆˆ FBZ in . 6 ECE 407 – Spring 2009 – Farhan Rana – Cornell University From Classical to Quantum Description Define two new operators:                    qP qM N iqU qNM qa qP qM N iqU qNM qa                  ˆ 2 ˆ 2 ˆ ˆ 2 ˆ 2 ˆ                      qaqa N qM iqP qaqa qNM qU           ˆˆ 2 ˆ ˆˆ 2 ˆ        ','ˆ,ˆ qqqaqa    The commutation relations are: Note the inverse expressions:       ','ˆ,ˆ qqN i qPqU   ECE 407 – Spring 2009 – Farhan Rana – Cornell University From Classical to Quantum Description Use the expressions: in the Hamiltonian operator: to get:               FBZ in 2 1 ˆˆˆ q qaqaqH                      qaqa N qM iqP qaqa qNM qU           ˆˆ 2 ˆ ˆˆ 2 ˆ                   FBZ in 2 ,ˆ,ˆ 2 ˆ 2 ˆ q tqUtqUq NM qPqP M N H    7 ECE 407 – Spring 2009 – Farhan Rana – Cornell University From Classical to Quantum Description               FBZ in 2 1 ˆˆˆ q qaqaqH     The final answer: and the commutation relations tell us that: 1) The Hamiltonians of different lattice wave modes are uncoupled 2) The Hamiltonian of each lattice mode resembles that of a simple harmonic oscillator      1ˆ,ˆ  qaqa                FBZ in . FBZ in . ˆˆ 2 ˆˆ q Rqi q Rqi j j j eqaqa qNM eqURu          Finally, the atomic displacements can be expanded in terms of the phonon creation and destruction operators ECE 407 – Spring 2009 – Farhan Rana – Cornell University What are Phonons? Consider the Hamiltonian of just a single lattice wave mode:               2 1 ˆˆˆ qaqaqH    In analogy to the simple harmonic oscillator, its eigenstates, and the corresponding eigenenergies, must be of the form:  .............,,,nn qq 3210 where          qqqq nnqnqaqaqnH                    2 1 2 1 ˆˆˆ  This eigenstate corresponds to phonons in the lattice wave mode • A phonon corresponds to the minimum amount by which the energy of a lattice wave mode can be increased or decreased – it is the quantum of lattice wave energy • A lattice wave mode with phonons means the total energy of the lattice wave above the ground state energy of is • The ground state energy is not zero but equals and corresponds to quantum fluctuations of atoms around their equilibrium positions (but no phonons)   2q     qnq      2q    qn  qn 