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Date Created: 09/19/15
Review of Quantum Mechanics 21 States and Operators A quantum mechanical system is de ned by a Hilbert space H whose vectors wgt are associated with the states of the system A state of the system is represented by the set of vectors 6 There are linear operators 0 which act on this Hilbert space These operators correspond to physical observables Finally there is an inner 20gt Xgt A 7gt gives a complete description of a system through the expectation 7gt assuming that l gt is normalized so that ltwlwgt 1 which would product which assigns a complex number ltXl gt to any pair of states state vector values ltw 0i be the average values of the corresponding physical observables if we could measure them on an in nite collection of identical systems each in the state The adjoint OT of an operator is de ned according to ltgtltl OW 040 W 21gt In other words the inner product between Xgt and Ol gt is the same as that between Ollxgt and An Hermitian operator satis es 0 OT 22 Chapter 2 Review of Quantum Mechanics 8 while a unitary operator satis es 00T 010 1 23 lf 0 is Herrnitian7 then em 24 is unitary Given an Herrnitian operator7 07 its eigenstates are orthogonal7 ltX 0b lt Agt XltX Agt 25 For A 31 X ltX If there are 71 states with the same eigenvalue7 then7 within the subspace spanned by Agt 0 26 these states7 we can pick a set of n mutually orthogonal states Hence7 we can use the eigenstates Agt as a basis for Hilbert space Any state 1 can be expanded in the basis given by the eigenstates of O 72 ZCAW 27 A with A particularly important operator is the Harniltonian7 or the total energy7 which we will denote by H Schrodinger7s equation tells us that H determines how a state of the system will evolve in time 3 malt legt 29 If the Hamiltonian is independent of tirne7 then we can de ne energy eigenstates7 210 Chapter 2 Review of Quantum Mechanics 9 which evolve in time according to Et Etgt w E0gt 211 An arbitrary state can be expanded in the basis of energy eigenstates Wiza 212 It will evolve according to Ejt MW 20157117 For example consider a particle in 1D The Hilbert space consists of all continuous 117 213 complex valued functions The position operator i and momentum operator 1 are de ned by 92 1W E 961ng p we 2 7mg W 214 The position eigenfunctions 6xia a6xia 215 are Dirac delta functions which are not continuous functions but can be de ned as the limit of continuous functions 2 1 an 6 l 37 216 z 6 lt gt The momentum eigenfunctions are plane waves 423 em hk em 2 17 h 39 Expanding a state in the basis of momentum eigenstates is the same as taking its Fourier transform 7112 fodk 713k 21s Chapter 2 Review of Quantum Mechanics where the Fourier coef cients are given by 123k czzw If the particle is free 2 2 then momentum eigenstates are also energy eigenstates 2m Heikm If a particle is in a Gaussian wavepacket at the origin at time t 0 1 72 a ea 1149670 Then at time t it will be in the state 2 4M 1 2 2 39 1 2k 1 61km 1 00 a 7Lt m mdk e 2m 6 22 Density and Current Multiplying the free particle Schrodinger equation by 7 wwhgw igu Z and subtracting the complex conjugate of this equation we nd gum 6 WW 7 W w This is in the form of a continuity equation The density and current are given by pww 219 220 221 222 223 224 225 226 Chapter 2 Review of Quantum Mechanics 11 5 E m 7 W t 227 2m The current carried by a plane wave state is 7 h a 1 39 k 228 7 2m 27f3 23 6function scatterer 2 12 62 H 7 V6 229 m 6952 96 6 Re ll if z lt 0 M96 230 Tell if z gt 0 T 7 1 7 m2 lt gt R 231 1 7 1M There is a bound state at 232 24 Particle in a BOX Particle in a 1D region of length L 7 12 62 H 7 233 2m 3x2 714 A617 Be ik 234 has energy E h2k22m 7M0 7L 0 Therefore7 714 A sin 235 Chapter 2 Review of Quantum Mechanics for integer n Allowed energies hzwznz n 2mL2 In a 3D box of side L7 the energy eigenfunctions are Mm A sin sin sin and the allowed energies are h27r2 nm 25 Harmonic Oscillator 7 12 62 1 2 Writing in km7 f pkm14i xkm l47 1 2 2 H 7 5w lt10 z gt 57 fl 43971 Raising and lowering operators ai 15 Mafiam Hamiltonian and commutation relations The commutation relations7 Hal hwal 236 237 233 239 240 241 242 243 Chapter 2 Review of Quantum Mechanics 13 Ha ihwa 244 imply that there is a ladder of states HallE EhwaTEgt HalE E i hwalE 245 This ladder will continue down to negative energies which it can7t since the Hamil tonian is manifestly positive de nite unless there is an E0 2 0 such that alEO 0 246 Such a state has E0 Eta2 We label the states by their ala eigenvalues We have a complete set of H eigen states ln such that 1 HM m n 5 ln 247 and al l0 olt To get the normalization we write alln onln 1 Then lCan Wall n l 248 Hence alln xn 1ln 1 aln 71 249 26 Double Well V x 250 Chapter 2 Review of Quantum Mechanics 14 where 00 if gt 2a 2b Va 0 ifbltlxlltab Vb if lt b Symmetrical solutions Acosk m if ltb z 251 cosklxl 7 b if b lt lt a b with 2 V k k2 7 m2 0 252 h The allowed k7s are determined by the condition that 7a b 0 l 5 71 5 7r7kab 253 the continuity of Mm at b 7 cos kb 7 b A 254 and the continuity of Xix at b k tan 7T 7 1m k tan M 255 If k is imaginary7 cos 7 cosh and tan 7 239 tanh in the above equations Antisymmetrical solutions Asink x if ltb we 7 256 sgn cosklxl 7 b if b lt lt a b The allowed k7s are now determined by l 5 ltn gt7r7kab 257 kb 7 A M 258 sin k b Chapter 2 Review of Quantum Mechanics 15 1 k tan 7T 7 ha 7 k cot M 259 Suppose we have 71 wells Sequences of eigenstates7 classi ed according to their eigenvalues under translations between the wells 27 Spin The electron carries spin 12 The spin is described by a state in the Hilbert space algt 6H 260 spanned by the basis vectors Spin operators 1 0 1 51 2 1 0 1 0 72 5 y 2 2 0 1 1 0 51 261 2 0 i1 Coupling to an external magnetic eld Hint 791333 262 States of a spin in a magnetic eld in the 2 direction Hlgt 7u3lgt HH MBH 263 28 ManyParticle Hilbert Spaces Bosons Fermions When we have a system with many particles7 we must now specify the states of all of the particles If we have two distinguishable particles whose Hilbert spaces are Chapter 2 Review of Quantum Mechanics 16 spanned by the bases 21 264 and Ot2gt 265 Then the two particle Hilbert space is spanned by the set 2162 2 2391gt 042gt 266 Suppose that the two single particle Hilbert spaces are identical eg the two particles are in the same box Then the two particle Hilbert space is m E 1gt j2gt 267 If the particles are identical however we must be more careful 239jgt and j239gt must be physically the same state ie 27 em 2 268 Applying this relation twice implies that 27 52 m 269 so em i1 The former corresponds to bosons while the latter corresponds to fermions The two particle Hilbert spaces of bosons and fermions are respectively spanned by Zyjgt m 270 and 277 The n particle Hilbert spaces of bosons and fermions are respectively spanned by 2 7r m 271 177 ngt 272 Chapter 2 Review of Quantum Mechanics 17 and W1 7W7 273 EH 7r ln position space7 this means that a bosonic wavefunction must be completely sym metric wz1mjn wx1xjin 274 while a fermionic wavefunction must be completely antisymmetric wz1mjn 77m7jixn 275
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