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Chapter 11 Wave Mechanics Chapter 11 Wave Mechanics P. J. Grandinetti Chem. 4300 P. J. Grandinetti Chapter 11: Wave Mechanics Wave Function in Position & Wavenumber Representation In quantum mechanics, we associate wavenumber with particle momentum p⃗ = `k⃗ Remember the Fourier transform relation for waves? ∞ ∞ ; ù1 ; *ikx ; ù1 ; ikx a.k t/ = Ê Ψ(x t/e dx and Ψ(x t/ = Ê a.k t/e dk 2휋 −∞ 2휋 −∞ Let’s replace k with p in the Fourier expansions... P. J. Grandinetti Chapter 11: Wave Mechanics Wave Function in Position and Momentum Representation Wave function in position and momentum representations become ∞ ; ù1 ; *ipx_` : Φ(p t/ = Ê Ψ(x t/ e dx 2휋` −∞ and ∞ ; ù1 ; ipx_` ; Ψ(x t/ = Ê Φ(p t/ e dp 2휋` −∞ We’ll use these expressions to predict a particle’s properties from its wave function. P. J. Grandinetti Chapter 11: Wave Mechanics Operators in Quantum Mechanics P. J. Grandinetti Chapter 11: Wave Mechanics Position Operators, x̂ Mean particle position, or expectation value of x, is weighted average calculated from wave function ∞ ∞ ê ë < ; ; x.t/ = Ê x p.x/ dx = Ê x Ψ .x t)Ψ(x t/ dx −∞ −∞ «­­­­­¯­­­­­¬ p.x/ In QM, we define x̂ operator: x̂ Ψ(x; t/ = x Ψ(x; t/ x̂ operates on Ψ(x; t/ to give back position times wave function. In formalism of quantum mechanics, we write êx.t/ë as ∞ ∞ ∞ ê ë < ; ̂ ; < ; ; < ; ; x.t/ = Ê Ψ .x t/ x Ψ(x t/ dx = Ê Ψ .x t/ x Ψ(x t/ dx = Ê x Ψ .x t)Ψ(x t/ dx −∞ −∞ −∞ «­­­­­¯­­­­­¬ p.x/ P. J. Grandinetti Chapter 11: Wave Mechanics Position Operators, x̂ Other mean or expectation value quantities, such as êx2.t/ë, are written ∞ ∞ ∞ ê 2 ë < ; ̂2 ; < ; 2 ; 2 < ; ; x .t/ = Ê Ψ .x t/ x Ψ(x t/ dx = Ê Ψ .x t/ x Ψ(x t/ dx = Ê x Ψ .x t/ Ψ(x t/ dx −∞ −∞ −∞ «­­­­­­¯­­­­­­¬ p.x/ For any function of x we write ∞ ∞ ∞ ê ; ë < ; ̂ ; ; < ; ; ; ; < ; ; f .x t/ = Ê Ψ .x t/ f .x t/ Ψ(x t/ dx = Ê Ψ .x t/ f .x t/ Ψ(x t/ dx = Ê f .x t/ Ψ .x t/ Ψ(x t/ dx −∞ −∞ −∞ «­­­­­­¯­­­­­­¬ p.x/ As long as the operator f̂.x; t/ depends only on x and t, then f̂.x; t/Ψ(x; t/ = f .x; t)Ψ(x; t/ What’s the point of introducing these extra steps with x̂? P. J. Grandinetti Chapter 11: Wave Mechanics Momentum Operators, p̂ What about mean or expectation value of particle momentum? Using wave function in momentum basis Φ(p; t/ we calculate ∞ ∞ ∞ ê ë < ; ̂ ; < ; ; < ; ; p.t/ = Ê Φ .p t/ p Φ(p t/ dp = Ê Φ .p t/ p Φ(p t/ dp = Ê p Φ .p t/ Φ(p t/ dp −∞ −∞ −∞ «­­­­­­¯­­­­­­¬ p.p/ Here p̂ operates on Φ(p; t/ to give p̂ Φ(p; t/ = pΦ(p; t/ Similarly we have ∞ ê 2 ë < ; ̂ 2 ; p .t/ = Ê Φ .p t/ p Φ(p t/dp −∞ and ∞ ê ; ë < ; ̂ ; ; f .p t/ = Ê Φ .p t/ f .p t/ Φ(p t/dp −∞ Again, what’s the point of introducing these extra steps with p̂? Let’s consider this further. P. J. Grandinetti Chapter 11: Wave Mechanics Momentum Operators If p̂ operates on Ψ(x; t/ we should get same result as p̂ acting on Φ(p; t/ ∞ ê ë < ; ̂ ; ; ̂ ; p.t/ = Ê Ψ .x t/ p Ψ(x t/dx but what is p Ψ(x t/ =?? −∞ Use Fourier transform to expand Ψ(x; t/ in momentum representation ∞ ̂ ; ù1 ̂ ; ipx_` p Ψ(x t/ = Ê p Φ(p t/e dp 2휋` −∞ Take derivative of Fourier expansion for Ψ(x; t/ ∞ ; ù1 ; ipx_` if Ψ(x t/ = Ê Φ(p t/e dp then 2휋` −∞ 0 1 ( ) ( ) ) ∞ ip ∞ ; ù1 ; ipx_` i ù1 ; ipx_` i ̂ ; ) Ψ(x t/ = Ê ` Φ(p t/e dp = ` Ê pΦ(p t/e dp = ` p Ψ(x t/ x 2휋` −∞ 2휋` −∞ ( ) ) i Ψ(x; t/ = p̂ Ψ(x; t/ )x ` P. J. Grandinetti Chapter 11: Wave Mechanics Momentum Operators Rearrange and find that p̂Ψ(x; t/ is given by ) p̂ Ψ(x; t/ = *i` Ψ(x; t/ )x Identify momentum operator when applied to wave function in position basis as ) p̂ = *i` )x Operator notation allows us to work with Ψ(x; t/ basis only and obtain expectation values for both x̂ and p̂. Homework Determine position operator when applied to wave function in momentum basis. P. J. Grandinetti Chapter 11: Wave Mechanics Eigenfunctions and Eigenvalues Whenever operator acts on wave function and gives back numerical value times original wavefunction then that wave function is an eigenfunction for that operator and numerical value returned is called the eigenvalue. Af̂ .§/ = a f .§/ f .§/ is eigenfunction of  and a is eigenvalue. Is Ψ(x; t/ eigenfunction for x̂? x̂Ψ(x; t/ = xΨ(x; t/, Yes Is Φ(p; t/ eigenfunction for p̂? p̂Φ(p; t/ = pΦ(p; t/, Yes dΨ(x; t/ Is Ψ(x; t/ eigenfunction for p̂? p̂Ψ(x; t/ = *i` , No dx Is Φ(p; t/ eigenfunction for x̂? No. Prove this at home. P. J. Grandinetti Chapter 11: Wave Mechanics Operators in Quantum Mechanics must be linear Can only lead to linear differential equations involving Ψ ( ) ̂ ̂ ̂ : A c1Ψ1 + c2Ψ2 = c1AΨ1 + c2AΨ2 Nonlinear operations, such as take square root log cosine sine raise to a power–other than 0 or 1 are not allowed. P. J. Grandinetti Chapter 11: Wave Mechanics Uncertainty and Commutator Relations P. J. Grandinetti Chapter 11: Wave Mechanics Commutation Relations Operators may not commute, i.e., order that operators are applied matters. Example What is result when operator below is applied to a wave function? x̂p̂ * p̂x̂ = ? Hint: not zero ) Recall p̂ = *i` . Apply this difference operator to Ψ(x; t/ we find )x 0 1 ( ) )Ψ(x; t/ ) .x̂p̂ * p̂x̂/Ψ(x; t/ = x̂p̂Ψ(x; t/* p̂x̂Ψ(x; t/ = *i` x * *i` .x Ψ(x; t// )x )x 0 1 )Ψ(x; t/ )Ψ(x; t/ = *i`x * *i`Ψ(x; t/* i`x = i`Ψ(x; t/ )x )x .x̂p̂ * p̂x̂/Ψ(x; t/ = i`Ψ(x; t/ P. J. Grandinetti Chapter 11: Wave Mechanics Commutation Relations .x̂p̂ * p̂x̂/ Ψ(x; t/ = i`Ψ(x; t/ Such operator product differences occur often in quantum mechanics. So often it is called a commutator and given shorthand notation x̂p̂ * p̂x̂ = [x̂;p ̂] = i`; for all Ψ(x; t/ When [ ; B̂ ] =  B̂ * B̂  ≠ 0 we say that  and B̂ do not commute. Why are commutation relations useful? P. J. Grandinetti Chapter 11: Wave Mechanics Uncertainty and Commutator Relations From [x̂;p ̂] = i` we obtain the uncertainty relation: êΔxëêΔpë g `_2 Commutator gives us more general uncertainty relationships. Given [ ; B̂ ] = iC; then êΔAëêΔBë g êCë_2 where ù êΔAë = êA2ë * êAë2 This general relation, êΔAëêΔBë g êCë_2, quantifies ability to specify precisely and simultaneously two observables for  and B̂ . P. J. Grandinetti Chapter 11: Wave Mechanics Kinetic and Potential Energy Operators P. J. Grandinetti Chapter 11: Wave Mechanics Kinetic and Potential Energy Operators Kinetic energy operator in momentum and position bases are p̂ 2 `2 )2 K̂ = and K̂ = * 2m 2m )x2 Potential energy operator, V̂ .x/, depend on system under study. Solving Schrödinger Eq. is easy or impossible depending on V̂ .x/. Examples of some easy ones are... Í Electron trapped in 1D box of length L has potential energy operator V̂ .x/ = 0 if 0 f x f L; V̂ .x/ = ∞ if x < 0 and x < L This is called the infinite well potential. Í 휅 1D harmonic oscillator with force constant f has potential energy operator 1 V̂ .x/ = 휅 x̂2 2 f P. J. Grandinetti Chapter 11: Wave Mechanics Total Energy Operator ≡ Hamiltonian Operator Total energy operator is sum of kinetic and potential energy operators, p̂2 `2 )2 Ĥ = K̂ + V̂ = + V̂ .x/ = * + V̂ .x/ 2m 2m )x2 Ĥ is called Hamiltonian operator. Recall Schrödinger equation 4 5 `2 )2 ) E Ψ(x; t/ = * + V̂ .x/ Ψ(x; t/ = i` Ψ(x; t/ 2m )x2 )t «­­­­­­­­­­¯­­­­­­­­­­¬ Ĥ We can write Schrödinger equation more compactly as ) E Ψ(x; t/ = Ĥ Ψ(x; t/ = i` Ψ(x; t/ )t With this substitution we see that Ĥ is also given by ) Ĥ = i` )t P. J. Grandinetti Chapter 11: Wave Mechanics Solving the Schrödinger Equation P. J. Grandinetti Chapter 11: Wave Mechanics Solving the Schrödinger Equation If potential energy operator, V̂ .x/, depends only on position and not time then separation of variables can be used Ψ(x; t/ = .x/휙.t/ Substituting into Schrödinger equation gives 4 5 `2 )2 `2 )2 .x/ )Ψ(x; t/ )휙.t/ * + V̂ .x/ .x/휙.t/ = * 휙.t/ + 휙.t/V̂ .x/ .x/ = i` = i` .x/ 2m )x2 2m )x2 )t )t Dividing both sides by Ψ(x; t/ = .x/휙.t/ leads to 4 5 1 `2 )2 .x/ 1 )휙.t/ * + V̂ .x/ .x/ = i` = E .x/ 2m )x2 휙.t/ )t «­­­­­­­­­­­­­­­­­­­­­­¯­­­­­­­­­­­­­­­­­­­­­­¬ «­­­­¯­­­­¬ depends only on x depends on on t For equality to remain true for all x and t both sides must equal separation constant, E, which we’ll find is total energy and is time independent. P. J. Grandinetti Chapter 11: Wave Mechanics Solving the Schrödinger Equation This gives 2 uncoupled ODEs: d휙.t/ iE d2 .x/ 2m ( ) + 휙.t/ = 0 and + E * V̂ .x/ .x/ = 0 dt ` dx2 `2 ODE for 휙.t/ has trivial solution: 휙.t/ = e*iEt_` ODE for .x/ is called the time independent Schrödinger equation.
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