C CS Phys 191 Uncertainty principle Spin Algebra Fall 2005 10 11 05 Lecture 13 1 Readings Uncertainty principle Griffiths Introduction to QM Ch 3 4 Spin Griffiths Introduction to QM Ch 4 4 Liboff Introductory Quantum Mechanics Ch 11 2 Heisenberg Uncertainty Principle Question Is it possible to construct a quantum state of well defined position and momentum A relevant theorem to help answer the question Theorem Consider two operators A and B representing two physical quantities It is possible to construct a simultaneous eigenstate ab of both A and B iff A B 0 where A B A B B A is the commutator between A and B Proof i First we show that if A B 0 then simultaneous eigenstates exist Suppose a is a set of non degenerate eigenstates of A A a a a Now consider B A a a B a But B A A B from the commutator so A B a a B a So we conclude that B a is also an eigenstate of A with eigenvalue a Now if these eigenstates are non degenerate then must be a multiple of since there can only be one eigenstate with eigenvalue a Therefore B a a i e B a b a where b is a constant Thus b is an eigenvalue of B Therefore a is a simultaneous eigenstate of A and B ii Now we show the converse If A and B have simultaneous eigenstates they are diagonalized by the same transformation T Then T T ABT T T AT T T BT A D B D B D A D T T BT T T AT T T BAT whence we have AB BA So it the two operators have simultaneous eigenstates they commute So to answer the question of whether we can construct a state of well defined position AND momentum then we must see if x p 0 or not We will evaluate the commutator in the position representation i e in the continuous basis x where x x meaning the position operator is just the function x We previously derived p in this position representation as p h i x Let s first test this operator p p k x h i x on an test state h ikx h keikx p k x e i x Now we can explicitly calculate the commutator in this basis h x i x C CS Phys 191 Fall 2005 Lecture 13 1 Notice the commutator is itself an operator in this case one that is asking to operate on some function Let s apply it to a test function f x and see what happens h h h h f f f x x f x f x x f x x x f x x x ih f x i x i x x i x x i x x We see that the test function f x is irrelevant and we find x p ih 6 0 Therefore we can conclude that you cannot simultaneously know the position and momentum of a quantum state with certainty This is one statement of the Heisenberg Uncertainty Principle This is often stated quantitatively as x p h 2 where A 2 is the variance of operator A i e A A 2 Note that the variance is defined for a particular state Similar uncertainty relations hold between all pairs of non commuting observables In your homework this week you prove the general quantitative form of the uncertainty relation between noncommuting observables A and B 2 1 Uncertainty principle and two slit experiment The uncertainty principle is responsible for one of the basic features of the two slit experiment namely that if one can observe an interference pattern that one has no knowledge of which slit the particle went through while if one measures which slit was traversed then one looses the interference pattern Figure 1 shows such a two slit experiment for a beam of electrons incident from the left The interference pattern is observed on a scintillation screen on the right Now suppose we put a source of photons after the slits which will interact with the electrons and allow us to find out measure which slit each electron passed through In order for this information to be achieved the position of each electron has to be measured within d 2 i e y d 2 Now if the interference pattern is to be maintained then the uncertainty in electron momentum induced by the interaction with a photon py must be very much less than a value that would displace the electron from a maximum in the interference pattern to an adjacent minimum This condition is readlily obtained from trigonometric analysis in Figure 1 giving py px 2 But we know from the de Broglie relation that p h and the angle at which the first interference maximum occurs is given by the path length difference between the 2 slits just like diffraction analysis see Figure 2 Thus d sin d Hence we have py f rach2d C CS Phys 191 Fall 2005 Lecture 13 2 Figure 1 Two slit experiment So now we have derived two inequalities that need to be satisfied if we can both determine which slit the electrons passed through and maintain an interference pattern Combining these two inequalities leads to h y py 4 Looks good NO this is in direct contradiction to the Heisenberg uncertainty relation for position and momentum The numerical factor of is not meaningful quantitatively can be fixed by appropriate definition of the uncertainty So we have to conclude that it is not possible to both measure which slit was traversed and maintain the interference pattern 3 Spin 3 1 Physical qubits Now after this foray into the world of wave mechanics let s get back to our discussion of qubits it is in the title of the course after all How can we make a qubit in real life We need a quantum mechanical two level system such that we can 1 Initialize the qubit C CS Phys 191 Fall 2005 Lecture 13 3 Figure 2 Diffraction path length difference at first diffraction peak is d sin 2 Manipulate the qubit think gates 3 Measure the qubit There are many other important issues such as decoherence and entanglement but we ll mainly be focusing on these first three Examples of some possible 2 level systems are spins atoms photons superconducting loops Over the next few lectures we ll be discussing how to physically prepare measure and manipulate real qubit systems made from spins 3 2 Recall what is spin Elementary particles and composite particles carry an intrinsic angular momentum called spin For our purposes the most important particles are electrons and protons They each contain a little angular momentum vector that can point up or down The quantum mechanical spin state of an electron or proton is thus Therefore spins can be used as qubits with 0 1 How do we understand the details of spin We gave a brief overview of the history and role of classical thinking in the development of spin in lecture 2 Please …