Lecture 3. QUANTUM MECHANICS FOR SINGLE QUBIT SYSTEMS 1. Vectors and Operators in Quantum State Space

Transcription

1 Lecture 3. QUANTUM MECHANICS FOR SINGLE QUBIT SYSTEMS 1. Vectors and Operators in Quantum State Space The principles of quantum mechanics and their application to the description of single and multiple qubits systems will the subject of our following discussion. Now, we are ready to penetrate more deeply into the structure of quantum mechanics in order to see the place of the Schrödinger equation in this structure, and to study how quantum computers can be built. The most elegant and simple way to achieve this goal is going back to the original Dirac formulation of the principles of quantum mechanics in terms of an abstract vector space. We define the quantum state space or (abstract vector space) as a collection of vectors a >, b >,...which represent the states of the physical system. If a > and b > are vectors, then their sum, a > + b >, is also vector of the same space and a > + b >= b > + a >; a > +( b > + c >) = ( a > + b >) + c >. (1) These relationships express the principle of superposition of states in quantum mechanics. The vectors a > and λ a > (λ is an arbitrary complex number) are supposed to represent the same state. Equally with the original vector space, the conjugate, or Hermitian conjugate space is introduced, by < a = ( a >) +, < b = ( b >) +,... (2) The vectors < a,... are usually called Hermitian conjugate, and the sign (+) relates to the operation of Hermitian conjugation. Or, following to Dirac terminology, vectors a > and < a are called ket- and bra-vectors, respectively. The correspondence between these two spaces are established by the conditions a > + b > < a + < b ; λ a > λ < a. (3) For every pair of vectors, a >, b >, the scalar, or inner product, < b a >, is defined such that < b a >=< a b > ; < a a > 0; < b λ a >= λ < b a >; < a ( b > + c >) =< a b > + < a c >. (4) Two vectors a > and b > are said to be orthogonal if If < b a >= 0 (or < a b >= 0). (5) < a a >= 0, (6)

2 vector a > is said to be zero vector: according to the definition, this vector does not correspond to any physical state. We notice that in accordance with (4) the inner product is not commutative: the inner product of vectors a > and b > is equal not to the inner product of vectors b > and a > but its complex-conjugate value. Due to this, the distinction between the ket- and bra-vector spaces exists side by side with the correspondence between them giving by Eqs. (3). The next step in building of quantum state space is very natural: we can suspect that the vectors in this space can be associated with each other by some prescription, or mapping, or, as it is usually called, by operator. We will denote operators by capital letters: A, B,... The special class of operators that satisfy the conditions L( a > + b >) = L a > +L b >; L(λ a >) = λl a >; (7) is most important for us because all dynamical variables in quantum mechanics are represented by only such operators which are called linear operators. Just as numbers can be added and multiplied, it is also natural to define sums and products of operators by the relations: (L 1 + L 2 ) a >= L 1 a > +L 2 a >; (L 1 L 2 ) a >= L 1 (L 2 a >) (8) The last equation states that the operator L 1 L 2 acting on vector a > produces the same vector that would be obtained if we first let L 2 act on a > and then L 1 on the result of the previous operation. But whereas with numbers ab = ba, for operators, there is no need to yield the same result if they are applied in thereverse order. Hence, in general L 1 L 2 L 2 L 1. The difference [L 1, L 2 ] L 1 L 2 L 2 L 1 (9) is called commutator of operators L 1 and L 2. Acordingly, the relationship [L 1, L 2 ] = ic, (10) where C is operator or (complex) number, is said to be commutation relation. In analogous way the anti-commutator, and anti-commutation relation, [L 1, L 2 ] + L 1 L 2 + L 2 L 1 (11) [L 1, L 2 ] + = ic, (12) can be defined. On ocasion we will also deal with antilinear operators. These share the properties (7-8) with linear operators with one exception: L(λ a >) = λ L a >. (13) Let us look into the conjugate space, when in ket-space a linear operator L acts on vector a >, b >= L a >. (14)

3 Because of the correspondence (3), the connection between adjoint vectors < b and < a is automatically established which we will symbolically fix by the relation < b =< a L +. (15) By this we introduce the correspondence < b =< a L + b >= L a >, (16) in which we use the rule for the multiplication to the left in bra-space and define new operator L +. If such operator is introduced, we can easily prove the following properties: (PROBLEMS: they will be in test 4) < a L + b >= (< b L a >) ; (L + ) + = L; (λl a >) + = λ < a L + ; [(L 1 + L 2 ) a >] + =< a (L L + 2 ); (L 1 L 2 a >) + =< a L + 2 L + 1 ; (< a L 1 L 2 L 3 ) + = L + 3 L + 2 L + 1 a >; < a L 1 L 2 b > =< b L + 2 L + 1 a >; (17) It can happen that L + = L, then L is self-adjoined or Hermitian operator. For it < a L b >= (< b L a >). (18) Besides operators, representing physical variables, there are many other useful operators. Here is several of them: (i) zero operator O, (ii) identity, or unit operator I, O a >= 0 for all a >; (19) I a >= a > for all a >; (20) (iii) inverse operator L 1, introduced in such way that (iv) operator P, We notice that L 1 L 2 = I L 2 = L 1 ; (L 1 L 2 ) 1 = L 2 L 1 ; (21) P = a >< b ( a >< b ) + = b >< a. (22) P c >= a >< b c >=< b c > a >, (23) i.e. the scalar product < b c > determines the projection of vector c > on vector a >. The operator P = a >< b is often called the outer product of vectors a > and b >. Finally, we define a unitary operator U by means of the relations: UU + = U + U = I. (24)

4 This relations imply the existense of the inverse operator U 1 equal to We notice that for unitary operators U 1 = U + (25) < a U + U b >=< a b >. (26) In particular, if vectors a > and b > are orthogonal, i.e. < a b >= 0, (27) then < a U + U b >= 0, (28) i.e. vectors U a > and U b > are also orthogonal. Therefore, we conclude that the transformation of quantum states representing by the unitary operator preserves orthogonality. 2. Eigenvalues and Eigenvectors of Operators The equation L l >= l l > (29) defines the eigenvalue problem for operator L. The number l and the ketvector l > are called eigenvalue and eigenvector, respectively. The eigenvector l > is said to belong to the eigenvalue l. If more than one eigenvector belongs to the same eigenvalue, we speak of degenerate eigenvalues. We call L operator with discrete or continuous spectrum, depending of the nature of the range of definition of l. In the conjugate space the eigenvalue problem has the form < l L = l < l. (30) We can easily verify the following properties of Hermitian operators (PROBLEMS will be in test 4): (i) all eigenvalues of the Hermitian operators are real-valued; (ii) eigenvectors, belonging to different eigenvalues, are orthogonal, i.e. { < l l Cδl,l (discrete spectrum); >= Cδ(l l ) (continuous spectrum), (31) C being the normalization constant (usually, C=1); (iii) eigenvalues for original (29) and adjoint (30) eigenvalue problems for Hermitian operator coincide. If the set of eigenvectors satisfies the completeness conditions or l l >< l = I (32) dl l >< l = I, (33) it is called complete set. From these conditions we immediately deduce that the arbitrary vector Ψ > can be expanded as Ψ >= l l >< l Ψ > (34)

5 or Ψ >= dl l >< l Ψ >. (35) Bearing in mind the terminology of an ordinary linear algebra, we can say that the quantities < l Ψ > plays the role of the projection of vector (or state) Ψ > on the basis vectors l > which can be pictured as defining a set of axes which are perpendicular in the sense of (31). The projection of the vector onto the axes shows the relative contributions of each eigenstate l > to the whole state Ψ >, rather like the components of a classical vector in ordinary Euclidean space. As in Euclidean space, the choice of a particular set of orthogonal axes is somewhat arbitrary. Different choices define what are called different bases. For example, you could obtain a new set of axes by rotating any given set of axes through a fixed angle. Although the state vector does not change in this process, its projections onto the various axes do. Consequently, the same state vector can assume a superficially different appearance if it is expressed in a different coordinate basis. However, this picture should not be interpreted too literally because the complete mathematical pedigree of the quantum state space asserts that it is a complex linear vector space. The label complex signifies that the components of the state vector have lengths that are complex numbers. Unfortunately, it is impossible to draw a line on a piece of paper that truly has a complex-valued lengths, so the ordinary axes which are conventionally used for depicting vectors can only hint at the structure of an authentic quantum state space. Nevertheless, like ordinary vectors quantum state vectors are specified by a particular choice of basic vectors (eigenstates) and a particular set of complex numbers, corresponding to the amplitudes with which each eigenstate contributes to the complete state vector. A simple two-state system of our interest has, by definition, two eigenstates which can be naturally designated as 0 > and 1 >. Putting C = 1 in (31), we obtain for these states and < 0 0 >= 1, < 0 1 >= 0, < 1 0 >= 0, < 1 1 >= 1 (36) 0 >< 1 1 >= 0 >< 1 1 >= 0 >, 0 >< 1 0 >= 0 >< 1 0 >= 0 > 0 = 0, 1 >< 0 1 >= 1 >< 1 0 >= 0, 1 >< 0 0 >= 1 >< 0 0 >= 1 >. (37) Thus, operator 0 >< 1 transforms the state 1 > to 0 > and 0 > to 0 (nule-vector). On the contrary, operator 1 >< 0 transforms the state 0 > to 1 > and the state 1 > to 0. Naturally, the sum of this transforms, 0 >< >< 0, (38) exchanges the states 0 > and 1 >. It is easy to verify also that (PROB- LEMS will be in test 4) 0 >< >< 1 = I, (39)

6 i.e. the states vectors 0 > and 1 > forms the complete set in the context of Eq. (32). For two-state system, the expansion (34) is written as Ψ >= l >< l Ψ >= 0 >< 0 Ψ > + 1 >< 1 Ψ > c 0 0 > +c 1 1 >. l (40) This expansion expresses the principle of superposition as the ability for a two-state quantum system to exist in a blend of two allowed states, 0 > and 1 > simultaneously and represents the possible state of a qubit. Nothing like this is possible classically! To get a more intuitive feel for superposition, it is useful to picture the state (40) as a vector contained in a sphere called Bloch sphere. The angle this vector makes with the vertical (polar) axis is related to the relative contributions of the eigenstates 0 > and 1 > to the whole state. The angle through which the vector is rotated about the polar axis corresponds to the phase. The phase factors do not affect the relative contributions of the eigenstates to the whole state, but actually the state have different amplitudes due to different phase factors. Naturally, this is crucially important for the quantum interference phenomena. 3. Matrix Representation of State Vectors and Operators In quantum mechanics the projections < l Ψ > are called the representatives of state Ψ > in L-representation. A remarkable observation: if all projections of vector Ψ > in any representation are known, then this vector is completely defined in this representation! So, we arrive to the presentation of ket-vectors in the form of a column matrix Ψ >= < l 1 Ψ > < l 2 Ψ >. < l n Ψ > In the adjoint space this looks as follows: < Ψ = l (41) < Ψ l >< l (42) or and < Ψ = dl < Ψ l >< l (43) < Ψ = ( < Ψ l 1 > < Ψ l 2 >... < Ψ l n > ) = ( < l 1 Ψ > < l 2 Ψ >... < l n (44) In particular case of a two-state system we have ( ) < 0 Ψ > Ψ >= < 1 Ψ > (45) and < Ψ = ( < Ψ 0 > < Ψ 1 > ) = ( < 0 Ψ > < 1 Ψ > ). (46)

7 For the basis vectors 0 > and 1 >, this gives or 0 >= ( ) ( ) < 0 0 > 1 = ; 1 >= < 0 1 > 0 ( ) ( ) < 1 0 > 0 = < 1 1 > 1 (47) < 0 = ( < 0 0 > < 0 1 > ) = ( 1 0 ) ; < 1 = ( < 1 0 > < 1 1 > ) = ( 0 1 ) (48) For arbitrary state Ψ > given by Eq. (40), we have Ψ >= c 0 0 > +c 1 1 >= c 0 ( 1 0 ) + c 1 ( 0 1 ) = ( c0 c 1 ). (49) Let A is an arbitrary operator. By virtue of the completeness condition, we can write A = l,l l >< l A l >< l l,l A ll l >< l. (50) By this we introduce the quantity A ll < l A l >, (51) which is called matrix element of operator A in L-representation. For a particular case A = L, { lδl,l (discrete spectrum); A ll = L ll = lδ(l l (52) ) (continuous spectrum). The substitution of this into Eq. (50) yields L = l l l >< l, (53) that is any operator L can be represented as a linear combination of the projection operators l >< l which coefficients are the eigenvalues of L. We see that (i) operator A in its own representation is diagonal, and (ii) diagonal elements of matrix A coincide with the eigenvalues of operator A. Therefore, the problem of calculation of the eigenvalues of operator A given in matrix form is reduced to the transformation of this matrix to the diagonal form. Generally, the eigenvalues of an arbitrary operator A can be always found from the secular or characteristic equation, det(a ll λδ ll ) = 0. (54) This is the equation of the n-th degree in the unknown λ, and its roots are the eigenvalues of A. It is extremely important that equation (50) can be constructed not only on the single set of eigenvectors of someone operator L but also on two different sets connected with two vectors, say, L and M, in which case instead of (50) we can write A = l >< l A m >< m A lm l >< m. (55) l,m l,m

8 In one way or another, we see that any operator can be represented in matrix form, and conversely, any matrix represents some operator along the rule, prescribed by Eqs. (50) or (55). For two operators A and B we have or < l AB l >= < l A l >< l B l >= A ll B l l AB (56) l l < l AB l >= < l A l >< l B l > dl = dl A ll B l l AB, (57) i.e. the matrix elements of the products of operators are found in accordance with the rule of matrix multiplication. The transformation of an operator from L to M representation can be constructed as follows: A ll < l A l >= mm < l m >< m A m >< m l > mm S lm A mm S l m and or in operator form Here, we introduced the Hermitian matrix (58) A mm SlmA ll S l m, (59) ll A L = SA M S + = SA M S 1 ; A M = S + A L S = S 1 A L S. (60) S lm =< l m >=< m l > = S ml, (61) which is naturally called matrix of transformation from L- to M-representation, or shortly, transformation matrix. This matrix is associated with the transformation operator S = l,m S lm l >< m, (62) which is Hermitian (S + = S) by definition. proved that Moreover, it can be easily S + S = SS + = I, (63) i.e. transformation matrix is unitary, S 1 = S +. For real-valued matrix S (S lm : real), S + = S and Eq. (63) takes the form SS = S S = I, (64) which means, in matrix terminology, that the transformation from L- to M-representation is the transformation of rotation. When the operator relations (60) take place, A M is said to be obtained from A L by a similarity transformation. It is worth to notice that the characteristic equation (54) is invariant under this transformation. Really, we have det(a M λi) = det[s 1 (A L λi)s] = det S 1 det(a L λi) det S = det(a L λi). (65)

9 In this proof the general rule for determinants, and the property of the transformation matrix, det AB = det A det B, (66) det S = det S 1 = 1, (67) have been used. Hence the eigenvalues of A as defined by Eq. (54) are independent of the representation. 4. Eigenvalue Problem and Measurement of Observables In preceding lecture we made certain of an importance of the eigenvalue problem as a natural tool for the representation of states and operators, and for the transformations from one repressentation to another. With regard to quantum mechanics, this problem plays an exclusive role in view of the following assumptions: 1. Let the system is in the state l >, one of the eigenstates of operator L, and dynamical variable L is measured in this state. It is claimed that the result of this measurement will be exactly l. 2. The reverse is also true: if at the measurement of the observable L the value l is obtained with authenticity, then the system is in the state l > (in the degenerate case, in one of the states belonging to the eigenvalue l). Logically, the last sentence does not imply that the system was in the state l > before the measurement: it means only that the system is switched at the measurement to the state l > (by this the physical sense of an action of operator is defined). In other words, the measurement in a quantum system results in a projection of its state prior to measurement onto the state compatible with the measured value. From this it follows that only Hermitian operators can correspond to the measurable physical quantities, and that the eigenvectors of such operators form the complete sets. On the other hand, if some quantity can be measured, the set of eigenvalues of the corresponding operator is complete. Such quantities are called observables. These two assumptions are extremely important but not sufficient because they don t answer to the main question: what will happen if one attempts to measure an arbitrary physical quantity L, when the system is in arbitrary state Ψ >, and is such measurement possible at all? Quantum mechanics teach us that as a result of such measurement, as a maximum, will be obtained one of the eigenvalues of operator L, and in the case of discrete spectrum the probability of such event is equal to P (l) = < l Ψ > 2 < Ψ Ψ > Ψ l 2 < Ψ Ψ >. (68) In the case of continuous spectrum the result of measurement belongs to the interval (l, l + dl) with the probability P (l)dl = Ψ l 2 dl. (69) < Ψ Ψ >

10 The function Ψ l =< l Ψ > (70) is naturally called the probability amplitude. From this statement it immediately follows that an average or mathematical expectation of the observable L in the state Ψ > is determined by the relation < L >= < Ψ L Ψ > < Ψ Ψ >. (71) In particular case, when the state Ψ > is normalized to unity, we obtain the famous formula < L >=< Ψ L Ψ >. (72) Warning: It is necessary to distinct strictly the angular brackets on the left and on the right in Eqs. (52) and (53): on the left they are used for signing an average whereas on the right they denote bra- and ket-vectors, as usual. If the state of the system is expressed as a weighted sum of some eigenstates of the operator L, then Eq. (34) yields and, if the state Ψ > is normalized to unity, Ψ >= l a l l >, (73) a l =< l Ψ > (74) < Ψ Ψ >= l a l 2 = 1, (75) so that the square moduli of the expansion coefficients a l determines the probability of the state l > in the superposition (73). Now, let consider the application of these ideas to the measurement of the observable L in the state Ψ > which itself is the superposition of two other arbitrary states Ψ 1 > and Ψ 2 >, For such state, Eq. (68) gives Ψ >= Ψ 1 > + Ψ 2 >. (76) P (l) = < l Ψ 1 > + < l Ψ 2 > 2 < Ψ Ψ > or, in the case of the normalized state, Ψ 1l + Ψ 2l 2 < Ψ Ψ > (77) P (l) = Ψ 1l + Ψ 2l 2 (78) We notice that this probability is generally not equal to the sum of probabilities, Ψ 1l + Ψ 2l 2 Ψ 1l 2 + Ψ 2l 2. (79) So, instead of the classical Bayes rule for combining probabilities we obtain new rule which shows how to combine the probability amplitudes. Don t

11 add probabilities, add probability amplitudes -teach us quantum mechanics! It is often refer to as Feynman s rule, after the famous Caltech physicist Richard Feynman. The key point is that probability amplitudes, < l Ψ k >, unlike probabilities < l Ψ k > 2, are not just simple positive numbers: they have a considerable freedom to take on negative values. In principle, that is no problem because to get the probability we have to square the probability amplitudes as Feynman s rule (77) says. However, there is an essential constraint: the final probabilities that we calculate for each of the mutually exclusive states (events) must add up to one. So, as a result of Feynman s rule some probabilities get smaller up to zero while others must get bigger up to one. 5. Uncertainty Principle Let take two observable, A and B, with commutator [A, B] = ic (80) and define the uncertainties A and B as the positive square roots of the variances or dispersions, ( A) 2 =< (A < A >) 2 >=< A 2 > < A > 2 ; ( B) 2 =< (B < B >) 2 >=< B 2 > < B > 2 ; (81) For these quantities we can prove (see Appendix) that or ( ) 1 2 ( A) 2 ( B) 2 2 < C > (82) A B 1 2 < C > (83) This relation expresses in most general form the famous Heisenberg uncertainty principle. This principle incorporates both the existence of such states in which some quantities can not be measured simultaneously, and, in its mathematical form, the technique which may be used for the description of such states, that is the technique of non-commuting operators. This technique is nothing but quantum mechanics itself because the presence of non-commuting operators is its main peculiarity as compared with classic mechanics. That is why the Heisenberg uncertainty principle is said to acquire the significance of one of the most fundamental principles of quantum mechanics. The uncertainty principle shows that the world at heart is random. Moreover, we immediately suspect that the randomness of quantum mechanics is not like ordinary everyday (classical) randomness. The latter is based on the idea that nothing is really random. That somewhere there is a well-hidden archive of data and variables, which, if accessed, could be used to predict with certainty what would happen in every case. The book of nature is written once and for all. Every element of physical reality has its marching orders. Every particle in the experiment does what it does

12 by virtue of its hidden instruction sets, its intrinsic nature. Hence, any observed randomness reflects our lack of knowledge of the hidden archive. And of course we believe, following to Newton and Einstein, that the uncertainty can always be totally removed if we have sufficient knowledge about a physical system and that knowledge can be gained by suitable experiments. However, the Heisenberg uncertainty principle shows that the reality is not like that. No amount of experimentation will ever remove the uncertainty of the quantum world - nature is irreducibly random! That is one of the most difficult things to accept when one first encounters the quantum theory. It is not surprising that many hidden variables theories have been extensively investigated as a sourse of randomness in quantum mechanics, but all have far failed to account for experimental results. As we saw, at every encounter with a beam -splitter, a photon makes a random choice to be reflected or transmitted, regardless of its previous history. The fate of the photon at every beam-splitter is always uncertain: the uncertainty is inexhaustible. It follows then that even a single photon has the capacity to provide an endless stream of information. More generally, if the quantum theory indicates the universe is irreducibly random, then the universe is inexhaustible sourse of information, a bottomless reservoir of surprise! Those who claim that we will soon have all the answers, and hence the science will soon end, do mistake. Note that the Universe is irreducibly random but at the same time it is apparantly intelligible as yet, and its intelligibility is achived by endless putting and getting answers to yes/no questions. We already know that at a deeper level the answer to any yes/no question is determined not by bit but a qubit which is governed by probability amplitudes and Feynman s rule. The apparent intelligibility of the Universe is constructed, bit by bit, from entangled qubits, the smart dice of reality. Thus the Heisenberg uncertainty principle, which at first sight looks as if it places an absolute limit on the power of information process, in reality leads to an enormous growth of this power through the idea of using qubits instead of bits: a sin is turned into a virtue! It took the genius of theoretical physicist Richard Feynman to discover this. However, let us return to the Heisenberg s principle itself and for, the moment, look for the case of commuting operators when C = 0, and hence ( A)( B) = 0. (84) It means that in principle the observables A and B can be measured simultaneously. But in practice such measurements is possible only if the system is in the eigenstate which is common or simultaneous for both operators A and B. This leads to the thought that two commuting operators must have the common set of eigenfunctions, and inversely, if A and B are obervables with common eigenfunctions which form the complete set, then operators A and B commute. This statement can be rigorously proved (P.A.M. Dirac), and has thus the character of theorem. These statements enables us to define more precisely the concept of the simultaneous measurement of two (or more) observables. In general case of non-commuting operators such measurement is prohibited because the measurement of the first observable, generally speaking, violates the states of the system, which influences on the result of the measurement of the

13 second observable. And only if the system is in the state which is common for both variables this violation can be vanished, and in this sense two variables can be measured simultaneously. In fact, these two successive observations can be considered as one common observation of more complicated type which result is expressed by not one but two numbers. In turn, it gives us the basis to consider two (or more) commuting observables as a single one, the result of which at the measurement consist of two or more numbers. Accordingly, it is convenient to mark the eigenfunctions (state) of such operators using the corresponding set of the characters (labels or quantum numbers), i.e. to write ab >, abc > etc. For any system in quantum mechanics, it can be introduced such maximum set of commuting Hermitian operators, that the eigenvalues of this set define unambiguously any possible state of the system. Obviously, such set define the complete set of the physical variables which can be measured simultaneously. In this sense, the operators belonging to this entire complete set, possesses the common eigenvectors which can be naturally enumerated by their eigenvalues l, m,... So, we can denote such eigenvectors as l, m,... > subjected to the close relation l,m,... l, m,... >< l, m,... = I (85) which expresses the completeness of this set of eigenvectors. arbitrary state of the system, Ψ >, can be expanded as where and Ψ >= l,m,... Then an l, m,... >< l, m,... Ψ >, (86) l, m,.. > l > m >.. (87) L l, m,... >= l l, m,... > etc. (88) Futhermore, the operator L can be represented as [clarify: Eq. (53)] L = l,m,... l l, m,... >< l, m,.... (89) If the partial sum of all projection operators, which correspond to the same eigenvalue l, is denoted by P l, we may rewrite (88) in the form L = l lp l, l P l = I. (90) Here, the sums extend over all distinct eigenvalues of L. These equations define what is called spectral decomposition of the Hermitian operator L. If we have the entire complete set, and some operator commute with all observables from this set, then this operator is simply the function of the observables from complete set. For such function we may write f(l, M,...) = f(l, m,...)p l,m,... l,... l,m,... P l,m,... = I, (91)

14 where P l,m,... is the partial sum of all projection operators, which correspond to the subset of the eigenvalues l, m,... Finally, we mention that for any two operators A and B the equality in uncertainty relation is achieved in the state Ψ > which is determined by the following conditions (see Appendix): (A < A >) Ψ >= λ(b < B >) Ψ >; < Ψ [(A < A >), (B < B >)] + Ψ >= 0. (92) This state is called the state with minimum uncertainty. Appendix 1. Proof of uncertainty relation: Eq. (82) Let α = A < A >, β = B < B >, α + = α, β + = β. Then [α, β] = αβ βα = (A < A >)(B < B >) (B < B >)(A < A >) = AB A < B > < A > B+ < A >< B > (BA B < A > < B > A+ < B >< A >) = AB BA = [A, B] = ic; < α >= 0, < β >= 0 α = α < α >= α, β = β, < ( α) 2 >=< α 2 >, < ( β) 2 >=< β 2 >. Therefore, < ( α) 2 >< ( β) 2 >=< α 2 >< β 2 >=< Ψ α 2 Ψ >< Ψ β 2 Ψ >. (93) Designations: χ >= β Ψ >, < χ =< Ψ β, ϕ >= α Ψ >, < ϕ =< Ψ α. (94) In these designations Schwarz inequality: Hence, Transformation: Thus < ϕ χ >=< Ψ αβ Ψ >; < ϕ ϕ >=< Ψ α 2 Ψ >; < χ χ >=< Ψ β 2 Ψ >. (95) < ϕ ϕ >< χ χ > < ϕ χ > 2 (96) < ( α) 2 >< ( β) 2 > < Ψ αβ Ψ > 2. (97) αβ = 1 2 (αβ + βα + αβ βα) = 1 2 (αβ + βα) + i C. (98) 2 < ( α) 2 >< ( β) 2 > 1 4 < Ψ (αβ + βα) Ψ > < Ψ C Ψ > 2 4 = 1 4 < Ψ( αβ + βα) Ψ > < C > < C > 2. (99)

15 i.e. < ( α) 2 >< ( β) 2 > 1 4 < C > 2. (100) 2. State with min uncertainty: Eq. (97) shows directly that the equality in (96) is achieved at the condition ϕ >= λ χ > (101) or α Ψ >= λβ Ψ >. (102) Eq. (99) gives additional condition < Ψ (αβ + βα) Ψ >= 0. (103)