Intent. a little geometry of eigenvectors, two-dimensional example. classical analog of QM, briefly

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Gordon Hopkins
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1 Intent a little geometry of eigenvectors, two-dimensional example classical analog of QM, briefly review of the underpinnings of QM, emphasizing Hilbert space operators with some simplifying assumptions??? sketch of the QM of a particle moving on the x-axis, again emphasizing Hilbert space operators

2 "Eigenomety" T :V "V a linear transformation " an eigenvalue with eigenvector v " Tv)= "v L={ "v : " a complex number } " TL)=L L is an invariant line for T )

3 Example T = 5 "1 "1 5 $ " = 4 1 " = 6 eigenvalues; v 1= 1 $ $ 1 u 1= v 1 v 1 " 1 = $ 1 $ 1 u = v " v 1 1 = $ "1 v = 1 eigenvectors "1 $ unit eigenvectors L = k { " u k : " a complex number } are invariant lines

4 Diagonalization: change of basis/coordinate system W = 1 $ 1 "1 1 1 unitary W "1 =W * "1 W = 1 $ 1 1 "1 1 Wv = v W e k )= v k W "1 TW = 4 0 "1 = D or D=WTW 0 6 $ R= rotation through angle " = cos" sin" $ sin" cos" ) )

5 The Classical System 1 particle, 1 degree of freedom, v= dx dt <<<c interaction of particle with environment described by force Fx) or potential Vx) [Fx)" d dx Vx)] Examples 1. Fx) = -kx " Vx) = " kx dx = k x +C. Vx)= k x " Fx)= d dx " $ $ k x = k x 3. F moves particle in direction that decreases V 4. V minimum " stable equilibrium V maximum " unstable equilibrium

6 How does the state of the system evolve with time? state " xt),pt), where pt) = m vt) $ x p have precise, well-defined values at every instant, and measurement does not disturb the system Object: given F = m a, determine xt) pt). $ dx d t = Fx) m =" dv dx m Examples 1. Fx) = k or Vx) = k x " xt)= k m t + C 1 t+ C pt)= kt+mc. Fx) = kx with x = 0 v= v 0, when t = 0 " xt)= v 0 " sin k m t) pt)= v 0 " kmcos k m t)

7 How does energy enter the picture? Calculate work W = " x1 x Fx)dx in two different ways: x dv x dv = "Vx) x1 x = Vx 1 ) Vx ) 1. W = " x1 dx dx = W = " x 1. W = " x1 x m x m dv dt dx = " dv x 1 dx v dx dt dx = " v mvdv= mv = mv v 1 v1 Equating 1.. " mv +Vx ) mv 1 = +Vx 1) x 1 x arbitrary " E " mv mv 1 +Vx) is constant particle only moves in region where Vx) < E E Vx) "measures" speed slope of Vx) "measures" direction and magnitude of F Hamiltonian formulation: "H "p = p m "H "x = dv dx Hx,p)" p m +Vx) " dx dt = "p " Hx,p) dp dt =" x Hx,p)

8 The Quantum System Axiom 1. States = { unit vectors in Hilbert Space H } With c = 1 "x) c"x) represent the same state. embody all knowable information about the system x is just a dummy variable Axiom. Observables = { self adjoint operators on H } may be unbounded; may have no eigenvalues position: M x "x) = x "x) momentum: P"x) = ih d" dx h" x joule sec) product rule " M x P PM x = ihi Note Classical states and observables are "joint," and they both depend on time. Quantum states and observables are distinct entities, and observables do not depend on time.

9 Axiom 3. The strongest predictive statement about the measurement of observable A, when the system is in state ", is probalistic: Eϕ ) = expected value of measurement of A = A"," A has an eigenbasis { e i } with A e i = " i e i, i = 1,,3,.... " Ee i ) = Ae i,e i = " i " = " c i e i " E" ) = i=1 " " i c i c i = ",e i ) i=1 " Probmeasuring A yields specific " k ) = c k " Probmeasuring A yields some " i ) = 1 If A has no eigenvalues, it still has a spectral representation: "dr"), RS)s are projections in BH ) for all measurable subsets of, S, of R. f "H " RS)f = " S f. Probmeasure of A between a and b) = Ra,b)" Unpredictability is inherent in the measurement process. What does this say about "discreteness of possible measurement values.

10 An aside: Why is the momentum operator P"x) = ih d" dx? Bohr 1913) circular orbit model of hydrogen atom F = Q r xt),yt))=rcos"t),rsin"t)) a =" r Newtons nd Law " m" r = Q r " " = Q mr 3 " p= mq r " angular momentum = J = mq r J "quantized" " J = nh " r n = n h mq and electron can move from one state to another, emitting photon with frequency = 1 h E " E ) n m

11 de Broglie 194) electron represented by a wave p= hk k = angular frequency of wave)! orbit consists of integer periods " "r = n " k!! h r " "r = n" = "nh p mq! " r = Bohr radii!! momentum is encoded in the oscillations of the wave function????

12 Suppose state of system is "x)= eikx " L [0,"]), and that it is an eigenvector of the momentum operator, P. Then Pϕ) = " k ϕ and should be " k = p= hk with probability 1. P= ih d dx uniquely does the trick! prob p= hk ) = P"," $ = P" " dx 0 e ikx = $ "ih d ) e "ikx )dx 0 dx $ = "ihik eikx ) e "ikx )dx 0 = hk " " 1) dx 0 = hk " ") hk $ $

13 Axiom 4. "Collapse" of the wave function The act of measurement causes an uncontrollable, immediate change in the system. Suppose that the system is in state ". Measurement of A yields a value ". Immediately after measurement the system will be in a possibly new state, ". A has an eigenbasis { e i } with A e i = " i e i, i = 1,,3,... " " immediately after measurement system will be in state " with A " =$ "; i.e. " =e k and " = " k for some k. A has no eigenvalues and a<" <b " immediately after measurement system will be in state Ra,b)" Ra,b)" Following the measurement, the state of the system will evolve in the "usual" way. A significant time later, the system will probably no longer be in state ".

14 Another aside: uncertainty, commutativity simultaneity Some statistics O an event with possible outcomes {o k } and probo k ) = p k EVO) = expected value of O = p k "o k "O = rms deviation of O = " p k o k EVO)) easy calculation " "O= EVO ) [EVO)] For observable operator A and state vector ϕ, uncertainty is defined by "A) = A A, I), and EVA) = A",". calculation " "A= EVA ) [EVA)] ΔA is called the uncertainty in A in the state ϕ.

15 Uncertainty Principle: AB BA = ci " ΔA ΔB 0.5 c If uncertainty in measurements of A decrease, then uncertainty in measurements of B must increase.) Proof. 1) If A and B are self adjoint, then A = A EVA)I B = B EVB)I are self adjoint, also. ) AB BA = AB BA 3) A",A" =A) and B",B" =B) 4) [AB BA]"," = [AB BA]"," = AB"," BA"," = B",A" A",B" = B",A" B",A" = iim B",A" So [AB BA]"," = Im B",A" B",A". CBS " [AB BA]"," B" $ A" = B",B" $ A",A" 3) " [AB BA]"," $B$A " ci"," = c "," = c $B$A ΔA = 0 " ϕ must be eigenvector measurement certain.

16 Position-Momentum Uncertainty Relation M x P PM x = ih " "M x "P$ h Compatibility Theorem A and B observables) A and B are simultaneously measurable compatible) " A and B have common eigenbasis " A and B commute Proof. Heuristically, why should this be true? If A measured, then B measured, then A measured again, all in rapid succession with no significant time evolution, the second measurement of A is certain to coincide with the first one.

17 Wave-Particle Duality " µ x)= 0 x µ $ x =µ eigenvectors of position M x "x)= x"x) spatially localized " PARTICLE " x)= e ix/h $ = cos x $ ) +i*sin x ) e.v.s of momentum P="ih d h h dx periodic with period "h " WAVE Cant be both wave and particle at the same time. Think about uncertainty relation and compatibility theorem. Problems: eigenvalues continuously distributed over the all reals. For all real a and b with a"µ"b, the integral of the Dirac delta b function is defined to be " µ x)dx = 1. a " = "," = $ e ix/h dx = $ 1dx = " needs to be normalized, but I dont know how.

18 Axiom 5. Time evolution/hamiltonian/schrödinger eq. d" dt = 1 ih H" where H" = momentum) " + potential)" m = m 1 P "+VM x )" = h m d " dx +Vx)" H behaves the way its supposed to. For example, " 0 =1 " " t =1 for all t > 0. Proof. Note that d dt "," " = t t t t," + " t t, " t write inner t product in its integral form, use product rule, and remember that integration wrt to x is independent of t). d" t dt = i h H" t " d dt " t = d dt " t," t = i h " t," t $ = i h H" t," t " t,h" t + " t, i h " t = 0, since H is self adjoint. ) Therefore, " t is constant.

19 Operator formulation doesnt make QM any easier. discover or invent appropriate H for given system solve Schrödinger eq. subject to " =" 0 when t = 0!!!!! How can we ever know " 0? "Prepare the system." Measure some observable A. System will be forced into eigenstate " k for A: " 0 = k. System will evolve according S.s eq. until another measurement is made. energy eigenvalue equation/time-independent S. equation: h m d dx " x) $ k +Vx)) " $ k x)= E k " k x) state time-evolution equation/time-dependent S. equation: h m d dx " x) $ t +Vx)) " x) $ t =ih *t * " x) t solve eigenvalue equation " can write down general sol. to time-evolution equation, given " t 0)

20 Example: Vx) = 0.5kx harmonic oscillator) eigenvalues of H = E k =k + 1 )h" k = 0, 1,, 3,... and " = k/m eigenvectors of H = " k x) = complicated exponential function of x

21 Time-evolution operators: Ut)" 0 x)=" t x) different Ut) for each " 0 Ut + s) = Ut)Us) U0) = I " {Ut)} is 1-param group think Stones Theorem) substitute into S.s eq. " ih "Ut) "t 0 x)= HUt) 0 x) want true for all " 0, so want ih "Ut) = HUt) with U0)= I "t solution of this operator d.e. is Ut)= e "iht/h Ut)s are linear operators, but not self adjoint, so not observables Why? Ut) = power series in H " e.v.s of Ut) are e "ie nt/h, where E n s are e.v.s of H, and these exps are not real.)

22 Time-Energy Uncertainty Relation The evolution time of observable A, T A, is defined by "A t T A deva) t dt T A is the time which must elapse before the average of the values measured for A in a series of repeated measurements, EVA) t, changes or evolves enough to be noticeable over the intrinsic spread in these values, "A t. T A "H t " h Proof. First, note that d dt EVA) t = i h " t,ha AH)" t. Then apply the Heisenberg Uncertainty Principle. The more precisely energy is defined the smaller "H t ) the slower any observable will change noticeably the larger T A ). If observable exhibits rapid variation, then system cannot have well-defined energy. If A commutes with H, then both EVA) t and "A t are constant in time. In particular, "H t and EVH) t are always constant in time.

23 Example: measure energy of electron in a hydrogen atom H has eigenvalues R n Suppose electron in state with energy R n1 Eventually decays into state with energy R n n <n 1 ) Photon emitted with energy E photon = R n R n1 = "E electron Frequency of photon " E photon which can be measured Can determine n 1 and n Do for many identically prepared electrons " get many frequencies/energies and probabilities look like Axiom 3. R = Rydberg constant = R= m eq 4 h

A few principles of classical and quantum mechanics

A few principles of classical and quantum mechanics The classical approach: In classical mechanics, we usually (but not exclusively) solve Newton s nd law of motion relating the acceleration a of the system