References: 1. Cohen Tannoudji Chapter 5 2. Quantum Chemistry Chapter 3

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1 Lecture #6 Today s Program:. Harmonic oscillator imortance. Quantum mechanical harmonic oscillations of ethylene molecule 3. Harmonic oscillator quantum mechanical general treatment 4. Angular momentum, classical and quantum mechanical Questions you will by able to answer by the end of today s lecture. Give a few examles for imortant hysical systems which can be described by a harmonic oscillator Hamiltonian.. Exlain the behavior of the quantum mechanical harmonic oscillator. 3. Know how to derive the orbital angular momentum observable. 4. Know how to check if a given vector observable is in fact an angular momentum Math tools covered today. Taylor exansions. Hermite olynomials another examle of a function basis set (see CT comliments) 3. Angular momentum observables. References:. Cohen Tannoudji Chater 5. Quantum Chemistry Chater 3

2 Imortance of the harmonic oscillator in hysics: A system exerting a force which is roortional to the dislacement F=-kx is called a harmonic oscillator. H( x, ) k k = + x = x = kx x x m x note: in 3D relace the D differentiation oerator with the gradient oerator = xˆ + yˆ + zˆ x x y z The force in an harmonic oscillator oerates in the a direction which is oosite from the dislacement it is thus called: a restoring force. The great imortance of this articular otential is that any smooth otential that has a minimum behaves to a certain extent like a harmonic oscillator (rovided that we are considering small deviations from this minimum oint). The Taylor exansion of a smooth otential about it s minimum x is: ( ) V x V x V V x x x x O x x ( ) = ( ) + ( ) + ( ) + ( ) x x= x! x x= x = since V is a minimum at x near the minimum the otential has a arabolic shae. 3 Examles of hysical systems which can be described by a harmonic oscillator model include: vibrational motion of diatomic molecules mention the Leonard Jones 6- otential, vibration of atoms in a solid (honons) and others. Torsional oscillations of the ethylene molecule (monomer for the famous olyethylene used for food ackaging, adhesives, films, lastic bags ect.): C σ σ σ π z H This molecule s lowest energy configuration is lanar i.e. if we were to observe the molecule from the z direction the angle between H atoms attached to the different C

3 atoms would be equal to zero at equilibrium. The two tyes of bonds formed in this molecule can be characterized in accordance with their transformation under a reflection oeration about the lane of the molecule: sigma σ bonds which have an electron eigenfunction that is unchanged by this oeration (in fact it is unchanged by any rotation about the axis connecting any two atoms) and π bond which is anti-symmetric about the lane of the molecule. The angles between the atoms is. The otential is aroximated by the following exression: V ( ) = ( cos) V V ( ) - 3 note: V ( ) V ( nπ) V ( ) V ( π) = + such that two stable equilibrium oints can be identified at = = = =. Classical descrition: where the angles reresent the rotation of the CH grou with resect to a fixed lane bisecting the two carbon atoms.

4 The otential deends only on = and the equations of motion can be written as: d d I = V = V dt dt ( ) ( ) d d I = V =+ V dt dt ( ) ( ) where I is the moment of inertia of the CH grou about the axis of rotation. Combine the two equations, d dt d I dt ( ) + = d = V d ( ) The first equation describes a constant rotation of the molecule about the z axis, the second describes the torsional motion. Let us now examine motion for small deviations about the equilibria ositions. The otential is exanded as leading to an harmonic oscillation. V ( ) = ( cos ) V V where we have used the taylor exansion ( ) f f ( ) = f ( ) + f ( ) + =! + cos... d d 4V I dt dt I ω φ = 4V + = ( t) = Acos( t+ ) where, ω t = 4V I t

5 Quantum mechanical qualitative treatment: Two degenerate modes associated with each energy level with corresonding eigenfunctions u ( ) and u ( ) and ( π) and ( π) u u u u V ( ) V ( ) u u u u Because the eigenfunctions located in each well have an overla (tunneling) we will have in-fact 4 different eigenfunctions. V ( ) u + u ħ δ u u + ħ δ Where, - 3 u = u + u u = u u u = u u u = u + u ( ) + ( ) ( ) ( ) +

6 The hysical imlication is that there will be oscillatory motion at different frequencies associated with the torsional movement of this molecule. The high frequency motion will be at aroximately δ δ and π π. ωt with low frequency corresonding to the slitting energy Quantum Mechanical quantitative harmonic oscillator I] The system: V(x) E x x M M x II] The classical energy function of the system: m (, ) = + V ( x) H x H x = + mω x = E (, ) m Side: using the Hamilton equations one can solve for the classical trajectory: xt = x cos ωt ϕ ( ) M ( ) ( ) = ωsin ( ω ϕ) t x m t M III] Obtaining the QM Hamiltonian oerator: (, ) ˆ ( ˆ, ˆ) ˆ( ) Pˆ ħ m m x H x H X P = + V x = + mω x

7 IV] Identify energy eigenfunctions Reminder: Any state of the system needs to satisfy Schrodinger s equation. ˆ ˆ ˆ ħ H X, P ψ xt, = iħ ψ xt, + mω x ψ xt, i ψ xt, = ħ t m x t ( ) ( ) ( ) ( ) ( ) As usual we focus on finding the energy eigenfunctions in cases which involve Hamiltonians which do not have exlicit time deendence. ħ (, ) ( ) = ( ) + ω ( ) = ( ) Hˆ Xˆ Pˆ u x Eu x m x u x Eu x m x The energy eigenfunctions are: The general form is, /4 /4 mω mω x ħ u ( x) = e π ħ /4 3/4 mω mω x ħ 4 u ( x) = xe π ħ /4 /4 mω mω mω x ħ u ( x) = x e 4π ħ ħ mω x ħ n ( ) = ( ) ( ) u x e h x h x in called an hermite olynomial. See Mathematica rogram: HermiteH[n, x] gives the Hermite olynomial H n HxL.

8 The corresonding energy eigenvalues: Consequences: En = ħ ω n+ n=,,. The minimal energy is always greater than no classical analogue.. The eigenfunctions have a definite arity. The lowest energy eigenfunction is even. 3. Energy is quantized, energy sectrum is discrete no classical analogue, result of being bound. 4. Sacing between the energy levels is constant. Angular Momentum Classical angular momentum is a vector quantity defined as, L = r In quantum mechanics we can define a corresonding observable (Hermitian oerator): Lˆ = Rˆ Pˆ Exlicitly: ˆ cos ( ˆ ˆˆ ϕ Lx YPz ZPy) i sin ϕ = = ħ + tan ϕ ( ) ˆ ˆˆ ˆˆ sinϕ Ly = ZPx XPz = iħ cosϕ + tan ϕ Lˆ ( ˆˆ ˆˆ ħ z = XPy YPx) = i ϕ ˆ L = ħ + + tan sin ϕ which we call the orbital angular momentum since it has a classical equivalent. If one looks at the commutation relations of this oerator one finds, Lˆ, ˆ ˆ x L y = iħlz Lˆ, ˆ ˆ y L z = iħlx Lˆ, ˆ ˆ z L x = iħly In QM there is another very imortant quantity which is called sin. It turn out that this quantity also obeys the same commutation rules and is therefore also called an angular momentum oerator, sin angular momentum. The sin angular momentum has no classical equivalent.