MS310 Quantum Physical Chemistry Ch 6. Commuting and Noncommuting Operators and the Surprising Consequence of Entanglement Applied simple quantum mechanical framework in real experiment. (Stern-Gerlach experiment) - Noncommuting operators concerning position and monentum. ( Heisenberg uncertainty principle) Particle in a 3-D box and Quantum computers MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry 6.1 Commutation relations There are 2 observables a and b, corresponding operator We can think two cases. 1) Measurement A first, B after 2) Measurement B first, A after If ψn(x) is eigenfunction of operator A(no state change) Also, if ψn(x) is eigenfunction of operator B(no state change also) MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry If ψn(x) is eigenfunction of operator B Also, if ψn(x) is eigenfunction of operator B ψn(x) is eigenfunction of operator A and B both → result is independent of the order of measurement Otherwise, two results are different. Two operator A and B have a common set of eigenfunction → must satisfy the commutation relation (f(x) is arbitrary function) and if it satisfied, A and B commute. Notation : MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry ψn(x) is eigenfunction of operator A : no change after the measurement the observable a If satisfied : state ψn(x) is not change by the two measurement the observable a and b → ‘can measure simultaneously and exactly two observable a and b’ Ex) 6.1 Momentum and a) kinetic energy b) the total energy can be known simultaneously? Sol) Use the commutator a) MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Momentum and kinetic energy is commute. Therefore, momentum and kinetic energy can be known simultaneously. b) Therefore, we cannot be known the momentum and total energy simultaneously. MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry 6.2 The Stern-Gerlach experiment Consider the dipole in the inhomogeneous magnetic field. In this situation, dipole orient and deflect to the magnetic field.(parallel and antiparallel to the magnetic field) MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Stern and Gerlach did the experiment. condition : external magnetic field applied to the Ag beam Result : Ag beam split two beams. → 2 eigenvalues of measure the z-component of the magnetic momentum We write operator of measurement the z-component of the magnetic momentum as A, wavefunction of one spin as α, other spin as β. MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Cannot specify the value of c1 and c2. However, ratio of two beam is 1 by the individual measurement. Measure the direction of x-component of magnetic momentum of the beam of state α : ‘split’ 2 beams! (in this case, write the operator : B and wavefunction : γ, δ) MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Then, operator A and B commute? No. If two operator commute → eigenfunctions of 2 operators same → result of second measurement is only 1 state. Why? ‘after’ the first measurement, wave function collapse to only 1 measured state. Second measurement measures the ‘collapsed’ state, one of the eigenfunctions of the first measurement. If two operator commute, second measurement measures the eigenfunction of operator B, and result must be one state. However, result of second measurement also split to 2 beams. Therefore, measurement of z-component of magnetic moment and measurement of x-component of magnetic moment do not commute. Result : Ag atom doesn’t have well-defined values for both μz and μx simultaneously. MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry 6.2.1 The history of the Stern-Gerlach experiment Experiment did in 1921 Ag beam generation : oven in a vacuum chamber was collimated by 2 narrow slits of 0.03mm width Beam passed into inhomogeneous magnetic field 3.5cm and impinged on a glass plate. 1 hr operation in this experiment. How can see the Ag? → ‘sulfur’ Sulfur reacts to Ag and makes Ag2S. Ag2S : black, and it can see less than 10-7 mol of Ag → reason of successful experiment MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry 6.3 The Heisenberg uncertainty principle Heisenberg uncertainty principle : ‘cannot know simultaneously position and momentum of particle’ It starts that position and momentum do not commute. Wavefunction of free particle : Ψ(x,t)=Aexp[i(kx – ωt – φ)] Set φ=0 and t=0 : focus on spatial variation of ψ(x) We normalized wavefunction into finite interval [-L,L] Probability of x=x0 : P(x0)dx=ψ*(x0)ψ(x0)dx L → ∞ : probability approaches to 0! → no data of position It gives this result : if we know momentum exactly, position is completely unknown Similarly, if we know position exactly, momentum is completely unknown MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Consider the superposition of plane waves of very similar wave vectors See the case of m=10(21 waves superposition) MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Wave vector k0 : 7.00 x 10-10 m Case of 21 waves, peak of the probability : 0 , 3.14 x 10-10 m → range of probability exist decrease(wave packet) : probability localized into finite interval → uncertainty of position increase. Superposition of a lot of plane wave : cannot know exactly the wave vector of particle → ‘uncertainty’ of momentum More wave superposition occurs, uncertainty of particle decrease, but uncertainty of momentum increase! Consider the ∆k << k0, momentum of wavefunction is given by It means, range of momentum increase when m increase. Finally, we can obtain Heisenberg uncertainty principle MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Text p.88 MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Text p.89 MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Text p.89 MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry 6.4 The Heisenberg uncertainty principle expressed in terms of standard deviation Heisenberg uncertainty principle can be written in the form σx,σp : standard deviation of position and momentum This 4 values are defined by postulate 4. MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Consider the particle in a box MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry n=1 : minimum → uncertainty principle satisfied for all n Relative uncertainty in x and p when n→∞, uncertainty of position increases. → related to probability of finding particle is equal everywhere case of large n. However, uncertainty of momentum is independent to n. → uncertainty of momentum can be negligible. But, it is not enough : there are 2 p values when p2 determined. Solution : change the wavefunction as eigenfunctions of momentum operator MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry relative probability density of wave vector Momentum approaches to classical value when n increase! → relative uncertainty of momentum ‘decrease’ as n increase! MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry 6.5 A thought experiment using a particle in a 3-dimensional box We do thought experiment by these steps. 1) A particle in a box → know the wavefunction of particle 2) Insert barrier → tunneling probability decrease in middle region 3) Move apart : separate to 2 boxes → wavefunction represent by (each function satisfies the particle in a box and can assume a=b) : superposition state 4) Look in box(measurement one of the boxes) → we can see only particle is in the left or right : ‘rapidly decay of superposition state when measurement occurs’ MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry 6.6 Entangled states, teleportation, and quantum computer Consider the case in 6.5, the particle(single particle) is in the superposition state. Also, this wavefunction is not an eigenfunction of position. If two quantum particles are strongly coupled : entangled state Beam of photons is incident on transparent cystalline BaTiO3. → Only 2 direction of electric field vector of photon : Horizontal(H) and Vertical(V) : polarization state Probability of horizontal and vertical is same by the measurement, and ‘if polarization of first photon measured, the other polarization will be measured exactly!’ MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Wavefunction can be described by This wavefunction is not an eigenfunction of single particle operator, and measure the single particle have no meaning because this system is ‘entangled’ state. MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry How can use this result : ‘teleportation’ There are a pair of entangled photons, Alice has photon A and Bob has photon B. Consider the photon A is entangled to photon X. It means photon A and photon X is orthogonal. Photon B is entangled to photon A and photon B must be orthogonal to photon A. Therefore, state of photon B is same as photon X, the message of Alice : teleportation! MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry More interesting application : quantum computer Classical computer(our PC) : bit n bit memory : 000…0 to 111…1 : 2n state Quantum computer : qubit Use the superposition of different quantum state. In photon system, H and V can be correspond to 0 and 1 Superposition state → qubit n-qubit system : entangled 2n state In a bit, 2M state stored in length M. However, in a qubit, 2M state stored in M-qubit, only one superposition state! Therefore, 2M simultaneous calculation can be parallel in M-qubit quantum computer. If M=30, 1030 calculation can be parallel, and it expects the speed of calculation improve surprisingly. MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Summary The Heisenberg uncertainty principle limits the degree to which observables of noncommuting operators can be known simultaneously. The Stern-Gerlach experiment clearly demonstrates that the prediction of quantum mechanics is obeyed at the atomic level. Entanglement is the basis of both teleportation and quantum computing MS310 Quantum Physical Chemistry