All you need to know about QM for this course

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1 Introduction to Eleentary Particle Physics. Note 04 Page 1 of 9 All you need to know about QM for this course Ψ(q) State of particles is described by a coplex contiguous wave function Ψ(q) of soe coordinates q. Coordinates q could be (t,x) or (E,p), but not both, plus soe other coordinates (e.g., particle s spin projection). dp Ψ(q) dq dp Ψ(q) dq represents probability (or can be just proportional to probability, depending on the choice of noralization) of finding particle with coordinates q in interval dq (assuing that q is contiguous variable, like coordinate x or oentu p). Note: Ψ(q) is not uniquely defined, i.e. Ψ (q)ψ(q)e iα leads to the sae probabilities. Ψ(q) αψ 1 (q) βψ (q) If Ψ 1 (q) describes soe physical state 1 and Ψ (q) describes a physical state,ψ(q) αψ 1 (q) βψ (q) describes a new physical state of superposition of the two states 1 and. Ψ(q 1, q ) Ψ 1 (q 1 )Ψ (q ) If Ψ 1 (q 1 ) describes a particle 1, Ψ (q ) describes a particle, and these particles do not interact, then their coon wave function Ψ(q 1, q ) Ψ 1 (q 1 )Ψ (q ) Wave Function Evolution. Measureent. The notion of an experient (observation) is the key in the conventional interpretation of QM: only by conducting an experient one can localize a particle with probabilities calculated according to Ψ(q) see point. Before an experient is conducted, the particle is at neither coordinate (i.e., one should not associate the wave function with soe physical seared distribution of atter). The quantitative role of QM is to give deterinistic description of the evolution of Ψ(q), rather than in deterinistic predictions for the outcoe of experients. In general, QM can give only probabilistic predictions for the experient outcoe. Equation describing tie evolution of Ψ can often be written as i Hˆ, where Hˆ is soe operator acting on function. t Ψ Ψ Ψ Reeber that the Plank s constant ħ h/π 1 in the Natural Units. It was shown that the QM probabilistic description could not be reduced to soe deterinistic, but hidden paraeters.

2 Introduction to Eleentary Particle Physics. Note 04 Page of 9 Observable variable and Operator. Eigenstates. 1. Any easured physical paraeter g can be associated with soe linear operator ĝ, such that, if one knows the particle wave function Ψ(q), the average easured value g can be calculated according to the integral: g Ψ gˆ Ψdq, where Ψ is conjugated and transposed vector Ψ; while the spectru of allowed g-values is found fro the following (in general, differential) equation: gψ gψ.. This equation ay lead to a contiguous or discrete set of solutions of g and Ψ. Discrete solutions usually results fro bounding conditions, like: a. constraining a particle to be in a potential box of size a discrete energy levels 1 ~ n a b. requireent that Ψ(ϕπ)Ψ(ϕ) discrete values for angular oentu: 0, ±1, ±, States Ψ and Ψ n for which g g n are orthogonal: Ψ Ψ dq n 0 If there are a few wave functions with a coon g k, then any linear cobination of the is also a solution. And one can always ix such functions in such a way that their linear cobinations also becoe orthogonal to each other. A few operators for function Ψ(t,x) a) coordinates: x x, y y, z z b) oenta: pˆ, ˆ, ˆ x i py i pz i x y z c) full energy (Hailtonian): Ĥ i (i.e., it is Hailtonian that drives the wave function evolution). t d) kinetic energy (non-relativistic!): ˆ pˆ E x y z e) potential energy: V V( x, y, z) f) angular oentu: lˆ ˆ ˆ, ˆ ˆ ˆ, ˆ ˆ ˆ, ˆ ˆ ˆ ˆ z xpy ypx lx ypz zpy ly zpx xpz l lx ly lz or in polar coordinates (R,θ,ϕ): ˆ ˆ 1 1 lz i and l sinθ ϕ sin θ ϕ sinθ θ θ Note that coordinate operators (a) follow fro the definition of Ψ(t,x). Operators (b), (c), (f) can be derived by considering invariance of physics laws under translations in space (b), tie (c), and angular rotations (f). Operators (f) also can be easily obtained by substituting QM oentu operators for oenta in the classical echanics equations for angular oentu. Operators for kinetic and potential energies can also be obtained the sae way.

3 Introduction to Eleentary Particle Physics. Note 04 Page 3 of 9 In-class exercises: Show how a energy/oentu quantization arises for a particle constrained to be between two walls. Show how the angular oentu quantization arises fro requiring continuity of wave function in ϕ.

4 Introduction to Eleentary Particle Physics. Note 04 Page 4 of 9 Wave function noralization 1. Any general state of a particle can be represented as a linear cobination of eigenstates. It is very instructive to revoke a geoetrical analogy of a general unit vector being expressed in ters of base unit vectors. For a set of discrete eigenstates: Ψ ( q) c Ψ ( q) where P c c would define a probability of being in state, all probabilities adding up to 1: 1 For a set of contiguous eigentstates Ψ ( q) c( ν) Ψ ( q)d ν where dp ν ν ν ν ν c( ) d defines a probability for a particle to have in an interval d, the integral probability over all states being equal to 1: c( ) d 1 ν ν. Condition of wave function orthogonality and the total probability noralizations can be cobined in one equation as follows: Ψ Ψ dq δ n for discrete spectru states, and Ψ Ψ dq δ ( ν ν ) for contiguous spectru states v v n The forer conditions are obvious. Noralization of contiguous spectru wave functions on the δ-function insures interpretation of c(ν) coefficients as given above. Indeed, ΔP Δν Ψ ( q) Ψ( q) dq ( Ψ ( q) Ψ ( q) dq) v δ ( ν ν ) c c( ν ) dν v dq Δν v c ( ν ) c( ν ) dν dν ( ν ) c( ν ) dν dν Ψ ( q) c ( ν ) dν Δν c( ν ) Ψ ( q) dν v

5 Introduction to Eleentary Particle Physics. Note 04 Page 5 of 9 Siultaneously easurable variables Note that eigenstates for operators g and f do not necessarily coincide. In this case, we say that values g and f cannot be siultaneously and precisely deterined. For two physical paraeters to be siultaneously defined, their operators should be coutative, i.e. g f f g, or [ g, f ] g f f g 0 def Uncertainty principle Note that operators of x and p x do not coutate: [ x, pˆ ] ˆ ˆ x xpx pxx i. Therefore, they cannot be precisely known at the sae tie. This leads to the faous Heisenberg s uncertainty principle (197): Δx Δ p x ~1 This has a nuber of iplications: a particle constrained in space within dx, will have uncertainty in oentu of dp~1/dx a particle scattered so that its oentu changes by dp can resolve sallest spatial structures of dx~1/dp There is a siilarly-looking equation for Δt Δ E ~1, which, for exaple, iplies that a particle of ass M can pop up into existence for a short tie dt~1/m a particle with lifetie of τ would have its ass spread over dm range, where dm~1/τ More exaples What can and cannot be easured siultaneously (check coutator): YES: x, y, z coordinates YES: all oentu coponents (p x, p y, and p z ) and the kinetic energy NO: x and p x NO: tie and full energy YES: x, p y, and p z NO: L x and L y or any other pair ( [L x,l y ]il z ) YES: L z and L (and siilarly for x- and y-coponents) YES: L z and p z (and siilarly for x- and y-coponents) NO: L z and p x or L z and p y YES: L z and z (and siilarly for x- and y-coponents) YES: L, S, J, and J z, where vector J is a full angular oent JLS

6 Introduction to Eleentary Particle Physics. Note 04 Page 6 of 9 Hailtonian Eigenstates Note that Hailtonian plays the dual role: it drives the wave function evolution and is the operator of the full energy H Ψ EΨ. For an enclosed syste Hailtonian cannot depend on tie and it obviously coutates with itself. Therefore, the latter equation should have a set of stationary solutions Ψ with conserved energy values of E. Cobining the two equations, one can easily obtain the explicit for of tie dependence of Ψ functions: ˆ HΨ and ˆ EΨ i Ψ HΨ, t fro where: i Ψ EΨ, t iet which has factorized solutions: Ψ ( txyz,,, ) e ψ ( xyz,, ). iet Then, any arbitrary state can be written as: ( txyz,,, ) c e ψ ( xyz,, ). Ψ Moentu Eigenstates 1. Eigenstates for oentu operator can be found fro the corresponding differential equation: pˆ xψ ( xyz,, ) pxψ ( xyz,, ) i ψ ( xyz,, ) px ψ ( xyz,, ) x ipxx which has solutions: ψ p ( xyz,, ) e χ( yz, ). x x y z Generalizing for other coordinates: ψ ( r) Ce Ce p ip x p y p z ipr Noralization 1 on the δ-function dxdydz 3/ Ψp Ψp δ ( p p) results in C 1( π ).. Cobining points 19 and 0 gives a wave function of a free particle: 1 iet ( pr) Ψ(E,p) e 3/ π 3. A general wave function can be, thus, presented as a superposition of oentu eigenstates: 3 1 ipr 3 3 ψ ( r) c( p) ψ p ( r) d p c( p) e d p, where d p dp. 3/ xdpydpz ( π ) Therefore, ψ ( r) and c( p) are Fourier transforations of each other. ( ) 4. iet ( pr) For a free particle in a very large, but fixed size volue V, the wave functions ΨCe can be noralized with C 1. In this case the integrated probability of finding the particle in the full volue V equals to one: V Ψ dv 1 1 Here one should use the property : e i x - α dx πδ ( α)

7 Introduction to Eleentary Particle Physics. Note 04 Page 7 of 9 Angular Moentu Eigenstates 1. Solutions for the operator of l z can be easily found in polar coordinates:, l (,, ) (,, ), zψ lzψ or i Ψ R θϕ Ψ R θϕ ϕ iϕ which has solutions: Ψ( R, θϕ, ) e f( R, θ). Fro a natural boundary condition of Ψ(ϕπ)Ψ(ϕ), one can see that l z can be equal only to integer nubers.. Since l and l z operator coutate, there should be solutions with definite l and l z, or, for short, l and. These solutions can be written in factorized for (p-functions are not noralized): iϕ Ψ( R, θϕ, ) f ( R) p ( θ) e, where 0 p ( θ) ± 1 p ( θ) cos θ, p ( θ) sinθ 1 0 ± 1 ± p ( θ) 3cos θ 1, p ( θ) cosθsin θ, p ( θ) sin θ etc... For fixed l, there are l1 possible l z -projections fro l to l and Δ l z 1. 1 l

8 Introduction to Eleentary Particle Physics. Note 04 Page 8 of 9 Spin Syetry of equations with respect to rotations allows introducing a plain internal angular oentu S (called spin) even for point particles without requiring anything physically spinning inside. Coutation rules for angular oentu projection operators (that can be derived fro syetry principles alone) drive the fact that Δ l z or Δ S z ust be an integral nuber. So the total nuber of projections for particles with spin S is S1: -S, -S1, -S,, -1, 0, 1,,, S-1, S, where spin S can take values of only 0, ½, 1, 3/,, etc. Spin-½ particle z-projection eigenstates can be presented as: 1 0 Ψ up ϕ( x) Ψdown χ( x) 0 1 And a general wave function and its conjugated state becoe: χ1 ( q) * * Ψ ( q) and Ψ ( q) ( χ1( q), χ( q) ) χ( q) Spin operators are atrices. For spin-½ particles, the operators are: ˆ ˆ ˆ 1 0 i Sz Sx Sy i 0 Ferions and Bosons 1. It was shown that indistinguishable particles with half-integral spins should obey Feri-Dirac statistics, i.e. their wave functions should be anti-syetrical under perutations of any pair of such particles: (1 ) Ψ(q 1, q ) -Ψ(q, q 1 ). Such particles are called ferions. Siilarly, indistinguishable particles with integral spins should obey Bose-Einstein statistics, i.e. their wave functions should be syetrical under swapping any pair of such particles: (1 ) Ψ(q 1, q ) Ψ(q, q 1 ). Such particles are called bosons.. Two iportant ferion properties: i. It is easy to show that no two ferions can be in the sae states. Fro the opposite, let s assue that there are two ferions with exactly the sae wave functions Ψ 0. The cobined wave function of two particles is Ψ A Ψ 1 (q 1 )Ψ (q ) Ψ 0 (q 1 )Ψ 0 (q ). After switching two ferions, the new function is Ψ B Ψ 0 (q )Ψ 0 (q 1 ). The two functions are required to be different by sign; however, they are clearly identical. This can be true only if Ψ 0 0. ii. A coposite particle ade of even nuber of ferions ust be a boson, and a coposite particle ade of odd nuber of ferions ust be a ferion. Indeed, let s put two identical coposite particles next two each other and assue that the nuber of ferions in each of the is N. Switching one pair of ferions between two coposite particles, changes sign of the wave function. After switching all N pairs, we effectively switched the two coposite particles, and the new wave function sign is (-1) N. If N is even, the overall sign has not changed and, therefore, the coposite particles ust have been bosons. 3. Two iportant boson properties: i. There is no liit on the nuber of boson that can take exactly the sae states. ii. A coposite particle ade of bosons ust be a boson.

9 Introduction to Eleentary Particle Physics. Note 04 Page 9 of 9 Quantu Mechanics oddities to stir up your ind: Apparent paradoxes (counter-intuitive phenoena): wave-particle duality non-local effects (Eienstein-Podolsky-Rosen "paradox") entangled states and hidden paraeters, Bell's theore, Aspect's experient null easureent (Elitzur-Vaidan bob-testing thought experient) spin phenoenon More serious issues: role of "easureent/observer", Schrödinger's cat QM interpretations QM and General Relativity