Physical constants Planck's constant 6.581×10-l6eVs Velocity of light in vacuum 2.9979 1010 cm/s Fine structure constant a=e-/hc 1/137.04 Rest mass of the electron mc2 0.511MeV Mass of the proton Mc- 938.28MeV Bohr radius 2/me2 5.2918×10-9cm Bohr magneton eh/2mc 0.58×10-8eV//gauss Boltzmann constant 8.62×10-5eV/K 1eV 1.6×10-i2erg
Physical constants Planck’s constant 6.581 × 10−16 eV s Velocity of light in vacuum c 2.9979 × 1010 cm/s Fine structure constant α = e2/c 1/137.04 Rest mass of the electron mc2 0.511 MeV Mass of the proton Mc2 938.28 MeV Bohr radius 2/me2 5.2918 ×10−9 cm Bohr magneton e/2mc 0.58 × 10−8 eV/gauss Boltzmann constant k 8.62 × 10−5 eV/K 1 eV 1.6 × 10−12 erg
Basic formalism We summarize below some of the postulates and definitions basic to our formalism,and on these postulates.The formalism is pur the the physical concepts that will be discussed in the later chapters will be framed. 1.1 State vectors It is important to realize that the Quantum Theory is a linear theory in which the physical state of a system is described by a vector in a complex,linear vector space.This vector may represent a free particle or a particle bound in an atom or a particle interacting with avector in ordinary three-din sional space,following many ofthe same rules,except that it describesa very complicated physical system.We will be elaborating further on this in the following. The mathematical structure of a quantum mechanical system will be presented in terms of the notations developed by Dirac A physical state in this notation is described by a"ket"vector,)designated variously as la)or)or a ket with other appropriate symbols depending on the specific problem at hand.The kets can be complex.Their complex conjugates,)*are designated by which are called"bra"vectors.Thus,corresponding to every ket vector there is a bra vector These vectors are abstract quantities whose physical interpretation is derived through their so-called"representatives"in the coordinate or momentum space or in a space appropriate to the problem under consideration. The dimensionality of the vector space open for the moment.It can be finite,as will be the case when we encounter spin,which has a finite number of components along a preferred direction,or it can be infinite,as is the case of the discrete bound states of the hydrogen atom.Or,the dimensionality could be continuous(indenumerable)infinity,as for a free particle with momer tum that takes continuous values.A complex vector space with these properties is called a Hilbert space The kets have the same properties as a vector in a linear vector space.Some of the most important of these properties are given below: (i)la)and cla),where c isa complex number,describe the same state (ii)The bra vector corresponding to cla)will be c*(l
1 Basic formalism We summarize below some of the postulates and definitions basic to our formalism, and present some important results based on these postulates. The formalism is purely mathematical in nature with very little physics input, but it provides the structure within which the physical concepts that will be discussed in the later chapters will be framed. 1.1 State vectors It is important to realize that the Quantum Theory is a linear theory in which the physical state of a system is described by a vector in a complex, linear vector space. This vector may represent a free particle or a particle bound in an atom or a particle interacting with other particles or with external fields. It is much like a vector in ordinary three-dimensional space, following many of the same rules, except that it describes a very complicated physical system. We will be elaborating further on this in the following. The mathematical structure of a quantum mechanical system will be presented in terms of the notations developed by Dirac. A physical state in this notation is described by a “ket” vector, |, designated variously as |α or |ψ or a ket with other appropriate symbols depending on the specific problem at hand. The kets can be complex. Their complex conjugates, |∗, are designated by | which are called “bra” vectors. Thus, corresponding to every ket vector there is a bra vector. These vectors are abstract quantities whose physical interpretation is derived through their so-called “representatives” in the coordinate or momentum space or in a space appropriate to the problem under consideration. The dimensionality of the vector space is left open for the moment. It can be finite, as will be the case when we encounter spin, which has a finite number of components along a preferred direction, or it can be infinite, as is the case of the discrete bound states of the hydrogen atom. Or, the dimensionality could be continuous (indenumerable) infinity, as for a free particle with momentum that takes continuous values. A complex vector space with these properties is called a Hilbert space. The kets have the same properties as a vector in a linear vector space. Some of the most important of these properties are given below: (i) |α and c |α, where c is a complex number, describe the same state. (ii) The bra vector corresponding to c |α will be c∗ α|
2 Basic formalism (ii)The kets follow a linear superposition principle ala)+b18)=cly) (1.1) where a,b,and c are complex numbers.That is,a linear combination of states in a vector space is also a state in the same space. (iv)The"scalar product"or"inner product"of two states )and B)is defined as (Bla). (1.2) It is a complex number and not a vector.Its complex conjugate is given by (B1a)*=(aβ) (1.3 Hence (olg)is a real number. (v)Two states l)and B)are orthogonal if 1a)=0. (1.4 (vi)It is postulated that(ala≥0.One calls√aa可the"norm”of the state.fa state vector is normalized to unity then aa）=l. (1.5) If the norm vanishes,then la)=0,in which case l)is called a null vector (vii)The states)with n=1,2,....depending on the dimensionality,are called a set of basis kets or basis states if they span the linear vector space.That is,any arbitrary are called orthonormal states.Hence an arbitrary state)can be expressed in terms of the basis states an)as 1)=>an lan) (1.6 where.as stated earlier,the values taken by the index n depends on whether the space is finite-or infinite-dimensional or continuous.In the latter case the summation is replaced by an integral.If the lo)'s are orthonormal then am=(n).It is then postulated that a is the probability that the state )will be in state ) (viii)Astate vector may depend on time,in which case one writes it as ()),())etc In the following,except when necessary,we will suppress the possible dependence on time. (ix)The product la)1B),has no meaning unless it refers to two different vector spaces, eg one con onding to spin,the other to momentum;or,if a state consists of two (x)Since bra vectors are obtained through complex conjugation of the ket vectors,the above properties can be easily carried over to the bra vectors
2 Basic formalism (iii) The kets follow a linear superposition principle a |α + b |β = c |γ (1.1) where a, b, and c are complex numbers. That is, a linear combination of states in a vector space is also a state in the same space. (iv) The “scalar product” or “inner product” of two states |α and |β is defined as β| α. (1.2) It is a complex number and not a vector. Its complex conjugate is given by β| α ∗ = α| β. (1.3) Hence α| α is a real number. (v) Two states |α and |β are orthogonal if β| α = 0. (1.4) (vi) It is postulated that α| α ≥ 0. One calls √α| α the “norm” of the state |α. If a state vector is normalized to unity then α| α = 1. (1.5) If the norm vanishes, then |α = 0, in which case |α is called a null vector. (vii) The states |αn with n = 1, 2, ... , depending on the dimensionality, are called a set of basis kets or basis states if they span the linear vector space. That is, any arbitrary state in this space can be expressed as a linear combination (superposition) of them. The basis states are often taken to be of unit norm and orthogonal, in which case they are called orthonormal states. Hence an arbitrary state |φ can be expressed in terms of the basis states |αn as |φ = n an |αn (1.6) where, as stated earlier, the values taken by the index n depends on whether the space is finite- or infinite-dimensional or continuous. In the latter case the summation is replaced by an integral. If the |αn’s are orthonormal then an = αn |φ. It is then postulated that |an| 2 is the probability that the state |φ will be in state |αn. (viii) A state vector may depend on time, in which case one writes it as |α(t), |ψ(t), etc. In the following, except when necessary, we will suppress the possible dependence on time. (ix) The product |α |β, has no meaning unless it refers to two different vector spaces, e.g., one corresponding to spin, the other to momentum; or, if a state consists of two particles described by |α and |β respectively. (x) Since bra vectors are obtained through complex conjugation of the ket vectors, the above properties can be easily carried over to the bra vectors
1.2 Operators and physical observables 1.2 Operators and physical observables A physical observable,like energy or momentum,is described by a linear operator that has the following properties: (i)If A is an operator and l)is a ket vector then A la)=another ket vector. (1.7) Similarly,for an operator B, (aB=another bra vector (1.8) where B operates to the left (ii)An operator A is linear if,for example. A[A la)+uB)]=4 la)+AIB) (1.9 where and u are complex numbers.Typical examples of linear operators are deriva- on the right-hand side are replaced by their complex conjugates.In this case it is called an antilinear operator. If an operator acting on a function gives rise to the square of that function,for with such operators. (iii)A is a called a unit operator if,for any la), A la)=la), (1.10) in which case one writes A=1. (1.11) (iv)Aproduct operator.In other words,if and Bare operators then AB as well as BA are operators.However,they do not necessarily commute under multiplication,that is, AB≠BA (1.12) in general.The operators commute under addition,ie.,if A and B are two operators then A+B=B+4. (113)
3 1.2 Operators and physical observables 1.2 Operators and physical observables A physical observable, like energy or momentum, is described by a linear operator that has the following properties: (i) If A is an operator and |α is a ket vector then A |α = another ket vector. (1.7) Similarly, for an operator B, α| B = another bra vector (1.8) where B operates to the left (ii) An operator A is linear if, for example, A [λ |α + μ|β] = λA |α + μA |β (1.9) where λ and μ are complex numbers. Typical examples of linear operators are derivatives, matrices, etc. There is one exception to this rule, which we will come across in Chapter 27 which involves the so called time reversal operator where the coefficients on the right-hand side are replaced by their complex conjugates. In this case it is called an antilinear operator. If an operator acting on a function gives rise to the square of that function, for example, then it is called a nonlinear operator. In this book we will be not be dealing with such operators. (iii) A is a called a unit operator if, for any |α, A |α = |α, (1.10) in which case one writes A = 1. (1.11) (iv) A product of two operators is also an operator. In other words, if A and B are operators then AB as well as BA are operators. However, they do not necessarily commute under multiplication, that is, AB = BA (1.12) in general. The operators commute under addition, i.e., if A and B are two operators then A + B = B + A. (1.13)
4 Basic formalism They also exhibit associativity,i.e.,if A,B,and C are three operators then A+(B+C)=(A+B)+C (1.14) Similarly A(BC)=(AB)C. (v)B is called an inverse of the operator A if AB=BA =1. (1.15) in which case one writes B=A-1 (1.16 (vi)The quantity la)(Bl is called the"outer product"between states la)and IB).By multiplying it with a statey)one obtains [la)(BI]Y)=[(BIy)]la) (1.17) where on the right-hand side the order ofthe terms is reversed since(y)is anumber. The above relation implies that when la)(BI multiplies with a state vector it gives another state vector.A similar result holds for the bra vectors: (y [la)(BI]=[(y la)](BI (1.18) Thus la)(Bl acts as an operator. (vii)The"expectation"value,()of an operator 4 in the state la)is defined as (4)=(alAla). (1.19) 1.3 Eigenstates (i)If the operation Al)gives rise to the same state vector,i.e..if Ala）=(constant)xla) (1.20) then we call la)an"eigenstate"of the operator 4,and the constant is called the"eigen- value"of.If)'s are eigenstates of with eigenvaluesassumed for convenienceto be discrete,then these states are generally designated as .They satisfy the equation Alan=an lan） (121) with=1,2....depending on the dimensionality of the system.In this case one may also call A an eigenoperator. (ii)If is an eigenstate of both operators A and B,such that A lan)=an lan),and Blan)=bn lan) (1.22)
4 Basic formalism They also exhibit associativity, i.e., if A, B, and C are three operators then A + (B + C) = (A + B) + C. (1.14) Similarly A (BC) = (AB) C. (v) B is called an inverse of the operator A if AB = BA = 1, (1.15) in which case one writes B = A−1. (1.16) (vi) The quantity |α β| is called the “outer product” between states |α and |β. By multiplying it with a state |γ one obtains [|α β|] γ = [β| γ ] |α (1.17) where on the right-hand side the order of the terms is reversed since β| γ is a number. The above relation implies that when |α β| multiplies with a state vector it gives another state vector. A similar result holds for the bra vectors: γ [|α β|] = [γ |α]β| . (1.18) Thus |α β| acts as an operator. (vii) The “expectation” value, A, of an operator A in the state |α is defined as A = α| A |α. (1.19) 1.3 Eigenstates (i) If the operation A |α gives rise to the same state vector, i.e., if A |α = (constant) × |α (1.20) then we call |α an “eigenstate” of the operator A, and the constant is called the “eigenvalue” ofA. If|α’s are eigenstates ofA with eigenvalues an, assumed for convenience to be discrete, then these states are generally designated as |an. They satisfy the equation A |an = an |an (1.21) with n = 1, 2, ... depending on the dimensionality of the system. In this case one may also call A an eigenoperator. (ii) If |αn is an eigenstate of both operators A and B, such that A |αn = an |αn, and B |αn = bn |αn (1.22)