note10 - Introduction to Quantum and Statistical Mechanics...

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Introduction to Quantum and Statistical Mechanics Handout 10 Quantum System with Many Degrees of Freedom Assigned Reading: Hagelstein, Senturia and Orlando, Chaps. 20, 23 and 24 10.1 Overview The real world is in 3D and has many particles, in contrast to the simplified quantum system we have dealt with until now. We will first treat the separable and nonseparable Hamiltonian in 3D with single particle, and then start a formalism where many particles (on the order of the Avogadro’s number) can be treated with a nearly continuum, preserving the quantum features. This is called the “density of states”, which evaluate how many available states can be fitted into a small range of energy. Remember however the physical measurements will always fix the system to one of its eigenstates, and therefore we are particularly interested in how to treat the transition from one state to the multiple states nearly in continuum. This is called the Fermi’s Golden Rule, which is very important in the quantum mechanical picture for the physical observables, such as mobility, radioactive decay and photovoltaic effects. 10.2 Separable and nonseparable Schrödinger equations When we treat the multi-dimensional problem, we first assure that the time-independent Schrödinger equation concerning the separation of time and spatial variables still applies. Identical to Eqs. (4.7) and (4.8), the eigenstates and eigenenergies work exactly the same way: ( 29 ( 29 ( 29 ( 29 t z y x t i t z y x z y x V t z y x m t z y x H , , , , , , ) , , ( , , , 2 , , , ˆ 2 2 ψ ψ ψ ψ = + - = (10.1) ) , , ( ) , , ( ) , , ( ) , , ( 2 2 2 z y x E z y x z y x V z y x m φ φ φ = + - (10.2) u / ) , , ( ) , , , ( iEt e z y x t z y x - = φ ψ (10.3) If the Hamiltonian is additively separable in the different coordinates, then we have ( 29 z H y H x H z y x H z y x ˆ ) ( ˆ ) ( ˆ ) , , ( ˆ + + = (10.4) Notice that since the kinetic term is separable in Cartesian coordinates (and by definition time-independent), the separability is basically determined by the potential V(x, y, z) . We can then assume that φ (x, y, z)= φ 1 (x) φ 2 (y) φ 3 (z) and obtain: 1
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z y x z z y y x x E E E E y E y y H y E y y H x E x x H + + = = = = ) ( ) ( ) ( ˆ ) ( ) ( ) ( ˆ ) ( ) ( ) ( ˆ 3 3 2 2 1 1 φ φ φ φ φ φ (10.5) ( Exercise 10.1 ) The kinetic energy operator 2 2 2 - m z is separable in the Cartesian coordinates. Can the same be applied to cylindrical and polar coordinates? 10.2.1 Free particle in 3D: separable in Cartesian coordinates For a free particle, we can see the plane wave in 3D works similarly in 1D ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ) ( exp exp exp exp , , 3 2 1 r r k i z ik y ik x ik z y x z y x z y x z z z φ φ φ φ φ = = = = (10.6) The dispersion relation is now: m k m k m k m k E E E E z y x z y x 2 2 2 2 2 2 2 2 2 2 2 2 u u u u = + + = + + = (10.7) Notice that this is one-to-one mapping to the momentum separation in classical mechanics, for the same reason as well. In each direction of x, y , and z , and in r , the uncertainty principle will hold.
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