Quantum Mechanics

Michael PustilnikSchool of Physics, Georgia Institute of Technology,

Atlanta, GA 30332

• Postulate 1For any quantum system there is an associated Hilbert space H. States of the system correspondto normalized vectors in H.

• Postulate 2For any classical observable A there is a corresponding Hermitian operator A = A† acting inH. Conversely, any Hermitian operator corresponds to some observable.

• Postulate 3A measurement of A yields one of the eigenvalues of the corresponding operator A.

• Postulate 4A measurement of A on many identical copies of the system, all in state |ψ〉, produces randomresults. The probability to find A = a is Pa = 〈ψ| 1a|ψ〉, where 1a is a projector onto the

subspace of eigenvectors of A corresponding to the eigenvalue a. Equivalently, probabilitydensity of A is P (A) = 〈ψ| δ

(A1− A

)|ψ〉.

• Postulate 5A measurement affects the state of the system: right after the measurement the system is inthe state |ψa〉 ∝ 1a|ψ〉, a normalized eigenvector of A corresponding to the eigenvalue a foundin the measurement. This change of the state as the result of the measurement is often referredto as the wave function collapse.

• Postulate 6Between the measurements, the state of the system evolves according to the Schrodinger equa-tion i~ d

dt|ψ〉 = H |ψ〉, where the Hamiltonian H is the operator corresponding to the energy of

the system.

Hilbert space

A linear vector space H is a collection of abstractvectors (often called ket-vectors) such that their linearcombinations also belong to H,

|ψ〉 , |ϕ〉 ∈ H ⇒ a |ψ〉+ b |ϕ〉 ∈ H, (1)

where a and b are arbitrary complex numbers. A sumof two (or more) vectors, multiplication of a vector by anumber, etc., have ‘natural’ properties, e.g.,

a |ψ〉+b |ψ〉 = (a+b) |ψ〉 , a |ψ〉+a |ϕ〉 = a(|ψ〉+a |ϕ〉

).

Any linear vector space contains a unique null -vector|null〉, such that for any |ψ〉 ∈ H

0 · |ψ〉 = |null〉 , |ψ〉+ |null〉 = |ψ〉 . (2)

A Hilbert space is a linear vector space endowed witha scalar product . Essentially, it is a rule that associatesa complex number with any two vectors from H,

|ϕ〉 , |φ〉 ∈ H −→ 〈ϕ|ψ〉 - complex number.

The scalar product 〈ϕ|ψ〉 is postulated to satisfy

〈ϕ|ψ〉 = 〈ψ|ϕ〉∗ (3)

〈ψ|ψ〉 ≥ 0 with 〈ψ|ψ〉 = 0 ⇔ |ψ〉 = |null〉 (4)

〈ϕ|(c1|ψ1〉+ c2|ψ2〉

)= c1〈ϕ|ψ1〉+ c2〈ϕ|ψ2〉 (5)

It can be shown that the Schwartz inequality,∣∣〈ϕ|ψ〉∣∣2 ≤ 〈ϕ|ϕ〉〈ψ|ψ〉, (6)

and the triangle inequality,√〈Φ|Φ〉 ≤

√〈ϕ|ϕ〉+

√〈ψ|ψ〉 for |Φ〉 = |ψ〉+ |ϕ〉 , (7)

2

follow from these postulates.We shall assume that the Hilbert space H comprises

only normalizable vectors with 〈ψ|ψ〉 < ∞ (this re-striction is important for infinitely dimensional Hilbertspaces).

A vector is called normalized if 〈ψ|ψ〉 = 1. Accordingto Postulate 1, all states of the system are described bysuch normalized vectors; every normalized vector corre-sponds to a state which, in principle, can be realized. Lin-ear combinations of such state vectors form the Hilbertspace of the system.

Basis

A set of vectors{|ψn〉

}, n = 1, . . . , N, is called lin-

early independent if the equation

n=N∑n=1

cn |ψn〉 = |null〉 (8)

has no solution for {cn} other than the trivial one (cn = 0for all n). Any linear vector space H is characterizedby its dimension NH = max{N}, the largest numberof vectors a linearly independent set can have. Thus,any set of NH linearly independent vectors serves as abasis for H: any |ψ〉 ∈ H can be written as a linearcombination of the basis vectors,

|ψ〉 =

n=NH∑n=1

ψn|ψn〉 . (9)

It can be shown that the coefficients ψn (often calledcomponents of vector |ψ〉 in the basis

{|ψn〉

}) in this

expansion are unique for a given basis set. Obviously,multiplying a vector by a number reduces to multiplyingall its components by that number. Adding two vectorsamounts to adding their components.

Given a linearly independent set of NH vectors, onecan use the so-called Gramm-Schmidt procedure to con-struct an orthonormal basis set

{|ψn〉

}, for which

〈ψm|ψn〉 = δm,n. (10)

From now on, we will only deal with such orthonormalbasis sets.

Components ψn of vector |ψ〉 in an orthonormal basis{|ψn〉

}have a particularly simple form ψn = 〈ψn|ψ〉, so

that

|ψ〉 =∑n

|ψn〉〈ψn|ψ〉. (11)

A scalar product of |ϕ〉 and |ψ〉 can be expressed via theircomponents in an orthonormal basis as

〈ϕ|ψ〉 =∑n

ϕ∗nψn. (12)

Obviously, 〈ϕ|ψ〉 given by Eq. (12) has all the propertieslisted in Eqs. (3)-(4) above.

Operators

Operator A acting in H maps the Hilbert space ontoitself: it is a rule that to any vector |ψ〉 ∈ H assigns someother vector |ψ′〉 ∈ H,

|ψ〉 A−−→ |ψ′〉 .

The vector |ψ′〉 is said to be the result of the action of A

on |ψ〉 and denoted as |Aψ〉 or A |ψ〉:

|ψ′〉 = |Aψ〉 = A |ψ〉 .

Our convention is that operators act on kets, never onbras. We are only interested in linear operators, suchthat

A(a |ψ〉+ b |ϕ〉

)= aA |ψ〉+ bA |ϕ〉 (13)

for any |ψ〉 , |ϕ〉 ∈ H.

Hermitian conjugate of operator A is such an oper-ator A† that

〈ψ| A |ψ〉∗ = 〈ψ| A†|ψ〉 for any |ψ〉 ∈ H, (14)

which implies that

〈ϕ| A |ψ〉∗ = 〈ψ| A†|ϕ〉 for any |ψ〉 , |ϕ〉 ∈ H. (15)

Hermitian operators satisfy A = A†. Hermitian op-erators have real eigenvalues (see below), hence the re-striction to Hermitian operators in Postulate 2.

A product of operators A and B is an operator C =AB that act as follows: C |ψ〉 = A|Bψ〉 (that is, B acts

first, then A acts on B|ψ〉). It is easy to verify that(AB)†

= B†A†. (16)

Note that C = AB is, in general, non-Hermitian even ifboth A and B are Hermitian.

The identity (or unity) operator 1 leaves vectors un-changed,

1|ψ〉 = |ψ〉 for any |ψ〉 ∈ H. (17)

Obviously, 1 = 1† and 12 = 1 (i.e., 1 is a projector).Comparison of Eqs. (11) and (17) shows that the identityoperator in the orthonormal basis {|ψn〉} can be writtenas

1 =

n=NH∑n=1

|ψn〉〈ψn| . (18)

Eq. (18) is known as the resolution of identity (a.k.a.the closure relation, a.k.a. the completeness relation).

Using Eq. (18), it is easy to obtain the representation

of A in terms of {|ψn〉},

A = 1A1 =∑m,n

|ψm〉Amn〈ψn| , Amn = 〈ψm| A |ψn〉 .

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The objects Amn form NH ×NH−matrix and are calledmatrix elements of the operator A. Obviously,

A = A† ⇔ Amn = A∗nm,

i.e., matrix elements of Hermitian operators form Hermi-tian matrices.

For C = AB we find Cmn =∑lAmlBln, i.e., the ma-

trix corresponding to the product of two operators is aproduct of their matrices. For |ψ′〉 = A |ψ〉 we haveψ′m =

∑nAmnψn, which can be viewed as a product

of the matrix representing A and a column-vector rep-resenting |ψ〉. Similarly, Eq. (12) above can be viewedas a (matrix) product of the row-vector representing 〈ϕ|,and the column-vector representing |ψ〉. Note that in thematrix language bras are Hermitian conjugates of kets,

|ψ〉 −→

ψ1

ψ2

. . .

, 〈ψ| −→(ψ∗1 ψ

∗2 . . .

)=

ψ1

ψ2

. . .

† .

Eigenvalue problem

Consider the equation

A |ψa〉 = a |ψa〉 , A = A†. (19)

The goal is to find all eigenvalues a for which Eq. (19)has a solution for the eigenvectors |ψa〉 ∈ H other than|ψa〉 = |null〉.

Obviously, all eigenvalues of Hermitian operators arereal,

Im a = 0. (20)

Eq. (20) guarantees that results of measurements are real,see Postulate 3. Moreover, it is easy to see that the eigen-vectors corresponding to different eigenvalues are orthog-onal,

〈ψa|ψb〉 ∝ δa,b, (21)

whereas any linear combination of eigenvectors corre-sponding to the same eigenvalue a is also an eigenvectorwith that eigenvalue. Accordingly, the eigenvectors cor-responding to the eigenvalue a form a linear vector spaceHa ⊂ H, a subspace of the Hilbert space of the system.

Let {|ψan〉

}, n = 1, . . . , Na ≤ NH

be an orthonormal basis for Ha. The eigenvalue a iscalled non-degenerate if the corresponding subspaceHa is one-dimensional, Na = 1. Otherwise, a is calledNa-fold degenerate.

The operator

1a =

n=Na∑n=1

|ψan〉〈ψan| (22)

is a projector onto Ha: 1a|ψ〉 ∈ Ha for any |ψ〉 ∈ H and

1a|ψ〉 = |null〉 if |ψ〉 is orthogonal to any vector in Ha.For a finite-dimensional Hilbert space (NH < ∞) one

can prove using the tools of Linear Algebra that∑a

Na = NH . (23)

Therefore, the union (set of sets) ∪a{|ψan〉

}forms the

orthonormal basis for the entire Hilbert space, and theidentity operator [see Eq. (17)] can be written in termsof the projectors (22) as

1 =∑a

1a. (24)

(Note that if a 6= a′, then the only element Ha and Ha′

share is |null〉; such spaces are called orthogonal.)We shall assume that Eq. (24) is applicable to

infinitely-dimensional Hilbert spaces as well: eigenvec-tors of Hermitian operators with discrete spectrum forma basis for the Hilbert space of the system. (This is in-deed so, provided that both the Hilbert space and theallowed operators are defined in a more careful mannerthan it is done here.) Obviously, operator A is diagonalin the basis of its own eigenvectors and is given by

A =∑a

a1a =∑a

n=Na∑n=1

|ψan〉 a 〈ψan| . (25)

Conversely, for any orthonormal basis set{|ψn〉

}there

exist Hermitian operators for which |ψn〉 are eigenvectors.Eq. (24) ensures that the probability introduced in

Postulate 4 is properly normalized,∑a

Pa = 〈ψ|∑a

1a|ψ〉 = 〈ψ|ψ〉 = 1. (26)

Commuting operators

Theorem : Operators A and B commute if and only ifthere exists an orthonormal set of NH vectors which areeigenvectors of both A and B, simultaneously.

Proof : The sufficient condition is obvious. The nec-essary condition is proved as follows: Let Ha be the sub-space of eigenvectors of A corresponding to the eigenvaluea. If [A, B] = 0, then B |φ〉 ∈ Ha for any |φ〉 ∈ Ha. Ac-

cordingly, one can consider an eigenvalue problem for Bin the restricted space,

B∣∣ψa,b⟩ = b

∣∣ψa,b⟩ , ∣∣ψa,b⟩ ∈ Ha.

Solution of this problem divides Ha into orthogonal sub-spaces Ha,b: Ha = ∪bHa,b. Here, by construction, Ha,b is

the space of simultaneous eigenvectors of A and B witheigenvalues a, b. If{∣∣ψa,bn ⟩}, n = 1, . . . , Na,b

4

is an orthonormal basis for Ha,b, then, obviously, theunion ∪a,b

{∣∣ψa,bn ⟩} is an orthonormal basis for the entireHilbert space. Q.E.D.

Implications: It is obvious that the above theoremcan be restated as

[A, B] = 0 ⇔ 1a1b = 1b1a = 1a,b. (27)

Therefore, according to Postulate 5, after one mea-sures A and then B, state of the system collapses to|ψ〉 ∈ Ha,b, one of the normalized eigenvectors of A and

B. According to Postulate 4, observables A and B inthis state are no longer random, but have definite values,a and b, respectively. Any subsequent measurement ofA or B, in any order, will not change the state of thesystem. Moreover, the product AB [and, more generally,

any function f(A, B)] is Hermitian, with |ψ〉 being thecorresponding eigenvector. In this sense, A and B notonly may simultaneously have definite (non-random)values, but can also be measured simultaneously.

Complete set of commuting operatorsIf Ha,b is one-dimensional (Na,b = 1), then knowledge ofA = a and B = b completely characterizes the state ofthe system: there is no degeneracy. If, however, Na,b > 1,

then there exists at least one additional operator C thatcommutes with both A and B but is functionally inde-pendent from them, so that the dimension Na,b,c of thespace Ha,b,c formed by the simultaneous eigenvectors ofall three commuting operators satisfies Na,b,c < Na,b. IfNa,b,c > 1, then there exists a fourth independent com-muting operator, and so on.

The number of independent commuting operators oneneeds to completely lift the degeneracy is often referredto as the number of degrees of freedom. Essentially, itis the number of measurements one needs to performin order to project the state of the system onto a one-dimensional subspace. The corresponding observables(A,B,C, . . .) are often called good quantum numbers,

and the operators A, B, C . . . are said to form a completeset of commuting operators.

Commuting operators and degeneracyIf [A, B] = 0, [A, C] = 0, but [B, C] 6= 0, then at least one

of the eigenvalues of A is degenerate. Indeed, let |ψ〉 be

a simultaneous eigenvector of A and B with eigenvaluesa and b, respectively, but not an eigenvector of C. (If all

eigenvectors of A and B were also eigenvectors of C, thenthe three operators would commute.) Then |ϕ〉 = C |ψ〉is also an eigenvector of A, with the same eigenvalue a,but it is not an eigenvector of B. Therefore, |ψ〉 and|ϕ〉 cannot be proportional to each other, i.e., the twovectors are linearly independent. This implies that a isdegenerate: Na ≥ 2.

Example is provided by the angular momentum: J2

commutes with any component Jα, but components donot commute with each other. Accordingly, eigenvalues

of J2 must be degenerate.

Non-commuting operators

If A and B do not commute, then a generic state |ψ〉will not be an eigenstate of either of these operators. Theuncertainty of the observable A in the state |ψ〉,

δA =√〈A2〉 − 〈A〉2 , 〈An〉 = 〈ψ|An|ψ〉 ,

can be also written as

δA =

√〈ψ| ∆ 2

A |ψ〉, ∆A = A− 〈A〉1,

or

δA2 = 〈ψA|ψA〉, |ψA〉 = ∆A|ψ〉 .

Application of the Schwartz inequality (6) to |ψA〉 andsimilarly defined |ψB〉 yields the Heisenberg’s uncertaintyrelation

δA δB ≥ 1

2

∣∣〈ψ|[A, B]|ψ〉∣∣. (28)

Importantly, the left-hand side of Eq. (28) does not dealwith a single quantum system, nor does it deal with mea-surements of A and B on the same system. Instead, it hasto do with statistics: Suppose we start with many iden-tical copies of the system, all in the state |ψ〉. We thenmeasure A on some of these copies, and B on some othercopies. The measurements produce two sets of randomresults, for A and B, characterized by the uncertaintiesδA and δB, respectively. Eq. (28) establishes the relationbetween the results of these independent experiments.

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Unitary transformations

It is obvious that a unitary transformation

|ψ〉 → |ψ′〉 = U |ψ〉 , U U† = 1, (29)

preserves the value of the scalar product for any two vec-tors,

〈ψ′|ϕ′〉 = 〈ψ|ϕ〉. (30)

In fact, it can be shown that any transformation withthis property must be unitary (Wigner theorem).

A passive transformation is regarded as a changeof variables and does not affect the values of observablequantities. This is possible if the transformation of vec-tors is accompanied by the transformation of operatorsaccording to

A→ A′ = U A U†, (31)

which guarantees that the probability P (A), see Pos-tulate 4, is not affected by the transformation. (Note

that 〈ψ′|A′|ϕ′〉 = 〈ψ|A|ϕ〉 , U AnU† = (A′)n, and that A

and A′ have the same eigenvalues.) Passive transforma-tions are often used to simplify the problem of findingthe eigenvalues of the operators of interest.

An active transformation describes change of thestate of the system. In this case the operators do notchange, but the expectation values do:

〈ψ′|A |ψ′〉 6= 〈ψ| A |ψ〉 . (32)

Alternatively, since

〈ψ′|A |ψ′〉 = 〈ψ| U†A U†|ψ〉 , (33)

one can view the active transformation as a change ofoperators only,

A→ U†A U†, (34)

with states unaffected. (When U is the operator of evo-lution in time, these two points of view correspond to theso-called Schrodinger and Heisenberg pictures.)

Continuous unitary transformations

Consider a single-parameter family of transformationsU(ξ) with U(0) = 1 and with the group property

U(ξ1 + ξ2) = U(ξ1)U(ξ2). (35)

It is easy to prove the Stone theorem: any transformationthat satisfies Eq. (35) can be written as

U(ξ) = e−iξK , (36)

where K = K† = i(dU/dξ

)ξ→0

is the generator of the

transformation. For small ξ we have U(ξ) = 1−iξK+. . .,

i.e., the generator describes an infinitesimal transforma-tion. The operator A(ξ) = U†(ξ)A U(ξ), see Eq. (34),obeys Heisenberg equation,

dA/dξ = i[K, A], (37)

which is a quantum counterpart of the classical Liouvilleequation (2.10).

Generators corresponding to translation and rota-tion transformations can be deduced from the corre-sponding classical expressions with the result

Ta = e−i(a·p)/~, Rθ = e−i(θ·J)/~, (38)

where we have set ξ = 1. It is easy to check that thetransformations (38) indeed describe translations and ro-tations:

T †a r Ta = r + a1, Ta|r〉 = |r + a〉 ,

R†θ rRθ = r + θ×r + θ2(. . .), Rθ|r〉 = |r + θ×r + . . .〉.

Note that Ta1+a2 = Ta1 Ta2 = Ta2 Ta1 for any a1 and a2,

whereas Rθ1+θ2 6= Rθ1Rθ2 6= Rθ2Rθ1 except for θ1 ‖ θ2,as expected for translations and rotations.

6

Classical vs Quantum

Hamiltonian formulation

Consider a classical particle of mass m in the presenceof velocity-independent potential V . In the Hamiltonianformulation of Classical Mechanics, state of the particleis specified by its position r and its momentum p, a.k.a.canonical variables. The dynamics is described by theHamilton (a.k.a. canonical) equations of motion

r = ∇pH, p = −∇rH, (2.1)

where the dot denotes the total derivative with respectto time and

H(r,p; t) =p2

2m+ V (r, t), (2.2)

is the Hamiltonian. Indeed, substitution of Eq. (2.2) intoEq. (2.1) yields the Newton’s equations

r = p/m ⇒ p = mr, p = −∇r V (r, t).

All of the Hamiltonian Mechanics can be expressed interms of the so-called Poisson brackets. The Pois-son bracket {f, g} of dynamical quantities f(r,p, t) andg(r,p, t) is defined as{

f, g}

= ∇r f ·∇p g −∇p f ·∇r g. (2.3)

The properties of the Poisson brackets are very similarto those of the commutators in Quantum Mechanics. Inparticular,

{f, g} = −{g, f},{f + g, h} = {f, h}+ {g, h},{f, gh} = {f, g}h+ h{f, g},{

f, {g, h}}

+{g, {h, f}

}+{h, {f, g}

}= 0.

Using Eqs. (2.1) and (2.3), it is easy to derive theLiouville equation of motion

df

dt=∂f

∂t+{f,H

}. (2.4)

Eq. (2.4) is valid for any dynamical quantity f(r,p, t).Substituting for f canonical variables, one recovers theHamilton equations (2.1). For f = H we find dH/dt =∂H/∂t, which expresses the conservation of energy: E =H = const if the Hamiltonian does not explicitly dependon time.

Canonical transformations

A transformation

r→ r′(r,p), p→ p′(r,p) (2.5)

is called canonical if it preserves the Poisson brackets ofthe canonical variables, i.e.,

{r′α, p′β} = {rα, pβ} = δα,β ,

{r′α, r′β} = {rα, rβ} = 0, {p′α, p′β} = {pα, pβ} = 0.

Depending on how the transformation affects observ-able quantities, one can distinguish between passive andactive transformations. For a passive transformationEq. (2.5) is regarded as merely a change of variables,while observables remain unchanged:

f(r,p) = f(r(r′,p′),p(r′,p′)

).

It can be shown that for a canonical transformation{f, g}′ = {f, g} for any f and g; here {...}′ is the Poissonbracket evaluated in the new variables. Therefore, boththe Liouville equation (2.4) and the Hamilton equations(2.1) retain their form.

On the contrary, an active transformation describeschange of the state of the system {r,p} → {r′,p′} accom-panied by change of observables,

f(r′,p′) 6= f(r,p).

Infinitesimal canonical transformations can be castin the form

r′ = r + µ∇pΦ, p′ = p− µ∇rΦ, (2.6)

where Φ(r,p) is the so-called generating function and µ isan infinitesimally small dimensionless parameter. Directcalculation shows that for the transformation (2.6)

{r′α, p′β} − {rα, pβ} ∝ µ2,

with similar relations for {r′α, r′β} and {p′α, p′β}. In otherwords, the Poisson brackets are preserved in the first or-der in µ, which is sufficient for our purpose.

For Φ = a · p, where a is a constant vector which hasunits of length, Eq. (2.6) describes a translation (shift)in space by µa,

r′ = r + µa, p′ = p. (2.7)

For Φ = θ ·J, where θ is a constant dimensionless vectorand J = r × p is the angular momentum, Eq. (2.6) de-scribes a rotation by the angle µ|θ| about the directionof θ,

r′ = r + µθ × r, p′ = p + µθ × r. (2.8)

It is often said that linear (angular) momentum generatestranslation (rotation) in space.

Continuous transformation can be viewed as a se-quence of a large number of infinitesimal transformations,controlled by dimensionless parameter ξ: r(0) → r(ξ)

7

with r(0) = r. Increase of ξ by dξ is an infinitesimaltransformation described by

r(ξ + dξ) = r(ξ) + dξ∇pΦ, p(ξ + dξ) = p(ξ)− dξ∇rΦ,

see Eq. (2.6), which can be written in the form of differ-ential equations

dr/dξ = ∇pΦ, dp/dξ = −∇rΦ. (2.9)

For active continuous transformation the observablef(r(ξ),p(ξ)

)evolves according to

df/dξ = {f,Φ}. (2.10)

Note that Eqs. (2.9) and (2.10) have the form ofthe Hamilton equations (2.1) and the Liouville equation(2.4), with ξ and Φ playing the parts of t and H. In thissense, the Hamiltonian generates changes of dynamicalquantities with time, just as linear and angular momentagenerate translations and rotations in space.

If {f,Φ} = 0, the quantity f is said to be invariantwith respect to the transformation generated by Φ. Forf = H, such invariance implies that Φ is independentof time, see Eq. (2.4). Thus, translational (rotational)invariance of H gives rise to the conservation of linear(angular) momentum.

Canonical quantization

The similarity of the classical Poisson brackets andquantum commutators suggests the correspondence be-

tween classical and quantum observables, the Dirac’squantization rule{

A,B}

= C ←→ [A, B] = i~ C. (2.11)

Although the rule (2.11) has its limitations, it works per-fectly well for sufficiently simple quantities, ensuring thatquantum results have correct classical limit.

Using Eq. (2.11), we find

{rα, rβ} = 0 ←→ [rα, rβ ] = 0

{pα, pβ} = 0 ←→ [pα, pβ ] = 0

{rα, rβ} = δα,β ←→ [rα, pβ ] = i~δα,β1

The rule Eq. (2.11) is applicable to the components of

the orbital angular momentum J = r × p, J = r × p aswell, e.g.,{

Jx, Jy}

= Jz ←→[Jx, Jy

]= i~Jz

(In this case there is no ambiguity in constructing the

operator J: just as it is the case in Classical Mechanics,r× p = − p× r, despite the non-commutativity of r andp.)

Classical-quantum analogy extends beyond the simi-larity of Poisson brackets and commutators. For ex-ample, classical Liouville equation (2.4) corresponds toquantum Heisenberg equation of motion, while the clas-sical canonical transformations have a quantum counter-part in the unitary transformations.