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Fundamentals Fundamentals of quantum mechanics Quantum Theory of Light and Matter Quantum mechanics revision Elementary description in terms of wavefunction ψ(x) |ψ(x)|2 : probability measuring particle at position x More generally: state vector |ψi (cf ~a) P Scalar product: ha|bi – in components i ai∗ bi . Physical properties relate to Hermitian operators Ô|ψi → |φi Paul Eastham Operator algebra defines theory Position and momentum [q̂, p̂] = i~ Fundamentals Fundamentals Fundamentals of quantum mechanics Fundamentals of quantum mechanics Measurements associated with Hermitian operators Ô Eigenstate of Ô|ei i = λi |ei i Arbitrary state= superposition of eigenstates of Ô Time evolution of state vector obeys i~ ∂ |ψi = Ĥ|ψ(t). ∂t |ψi = Usual Hamiltonian is classical energy with p, q → p̂, q̂ Ĥ = p̂2 2m X i ci |ei i. Measurement gives λi with probability |ci |2 . After measurement state is |ei i. + V (q̂). [Ô1 , Ô2 ] 6= 0 ⇒ eigenstates not orthogonal. measurement of O1 ⇒ superposition of e’states of O2 ⇒ uncertainty etc.. Origin of uncertainty relations Origin of uncertainty relations A general uncertainty relation A general uncertainty relation Many systems in some identical quantum state. Measure the values for Â, usual variance of the resulting data set Same for σA2 σB2 Useful if [A, B] = number, ic, same irrespective of state ⇒ σA2 σB2 = ha|aihb|bi = ha|aihb|bi ≥ |ha|bi|2 = hψ|(ÂB̂)|ψi . . . ⇒ha|bi − hb|ai = hψ[A, B]|ψi. σA2 = hψ|( − hAi)( − hA)|ψi = ha|ai σB2 z = ha|bi = hψ|( − hAi)(B̂ − hBi)|ψi = |z|2 ≥ (Imz)2 = z −z 2i ∗ 2 ≥ |ha|bi|2 = |z|2 ≥ (Imz)2 = z − z∗ 2i = 2 c2 4 Origin of uncertainty relations Heisenberg and Schrodinger Pictures A general uncertainty relation Heisenberg and Schrodinger Pictures c 2 ~ σp σq ≥ 2 σA σB ≥ Generalization of Heisenberg uncertainty relation About parallelism of eigenvectors; [A, B] = ic type operators can bound max angle < 90◦ Remember this is a bound not an equality “Minimum uncertainty states” saturate this bound This is not a statement that  (or B̂) alone is “imprecise” Asymmetrical uncertainty, e.g. σA2 < c 2 /4, “squeezed state” Heisenberg and Schrodinger Pictures Ladder operators for the harmonic oscillator Operator approach to harmonic oscillator Operator approach to harmonic oscillator EM field equivalent to a set of harmonic oscillators ∴ Ĥ = mω 2 2 p̂2 + q̂ , 2m 2 “Annihilation operator” â – â = r mω q̂ + i 2~ r 1 p̂, 2~mω ⇒ 1 Ĥ = ~ω a† a + 2 Suppose eigenstate of a† a w/eval n = |ni. √ â|ni ∝ |n − 1i = n|n − 1i √ ↠|ni ∝ |n + 1i = n + 1|ni. Lower bound on H ⇒ some state â|0i = 0. ∴ discrete ladder of integer n ⇒ energy ~ω(n + 1/2). ”Number states” or “Fock states” [q̂, p̂] = i~ ⇒ [â, ↠] = 1 Ladder operators for the harmonic oscillator Ladder operators for the harmonic oscillator Ladder operator proofs Operator approach to harmonic oscillator If ↠â|ni = n|ni, consider ↠â(â|ni) = (â↠− 1)â|ni ([a, a† ] = 1) † = â(â â|ni) − â|ni = (n − 1)(â|ni). So â|ni is an eigenstate with eigenvalue |n − 1i: â|ni = c|n − 1i. and mod-squaring this gives normalization |c|2 = n. Interested in position? Will need hq|ni. 1 q d â = √ +` . dq 2 ` So ground state obeys a|0i = 0 or 1 q d √ +` u0 (q) = 0 dq 2 ` 1/4 1 1 2 2 √ e−q /2` . ⇒ u0 (q) = π ` How do we get the excited state wavefunctions . . . ?