Erwin Schrodinger(1887-1961) Shared the Nobel Prize in 1933 with Paul Dirac for "the discovery of new productive forms of atomic theory." Refore 1925 auantum mechanies was an inconsistent col. observations differe of quantum mechanics were presented:Heisenberg pro posed a matrix mechanics of non-commuting observables. explaining discrete energy levels and quantum iumps while Schrodinger put forward wave mechanics.attributing a function of coordi. nates toa particle and a wave-like equat n governing its theory seemed(and seems)less abstract.Besides,it provided a clear mathematica description of the wave-particle duality,a concept actively advocated by Einstein. Therefore,many people favored his theory over Heisenberg's.As we explain in this chapter,both theories are equivalent.However,for many the Schrodinger equation remains the core of quantum mechanics. Erwin Schrodinger had a life with several restless periods in it.He was born and educated in Vienna,where he stayed till 1920,only interrupted by taking up his duty at the Italian front in the first wo orld war in the ars1920-1921he uccessivel accepted positions in Jena Stuttgart,Breslau,and fin ally Zurich,where he remaine for several years.The intense concentration that allowed him to develop his quantum theory he achieved during a Christmas stay in 1925 in a sanatorium in Arosa.But in 1927 he moved again,this time to Berlin.When the Nazis came to power in 1933 he decided to leave the Reich.and so he became itinerant again:Oxford,Edinburgh Graz,Rome,Ghe In 1940 he finally took a position in Dublin where he staye till his retirement in 1956. In this notation,A is an operator:it acts on the functionand it retums anothe function,the time derivative of.This operator H is called the Hamiltonian opera- tor,since it represents the total energy of the particle.This is best seen if we rewrite A in terms of the momentum operatorp=-ihar and position operator =r.We then see that A is identical to(1.3)but with the vectors r and p replaced by their operator analogues. Not only momentum and position.but any physical property can be expressed in terms of operators.For an arbitrary observable A,we can find the corresponding quantum mechan- ical operator A.The average value of this observable,sometimes called expectation value. in a given quantum state reads 4=d*). (1.6) being the complex conjugate of
5 1.2 Schrödinger equation Erwin Schrödinger (1887–1961) Shared the Nobel Prize in 1933 with Paul Dirac for “the discovery of new productive forms of atomic theory.” Before 1925, quantum mechanics was an inconsistent collection of results describing several otherwise unexplained observations. In 1925–1926 two different general theories of quantum mechanics were presented: Heisenberg proposed a matrix mechanics of non-commuting observables, explaining discrete energy levels and quantum jumps, while Schrödinger put forward wave mechanics, attributing a function of coordinates to a particle and a wave-like equation governing its evolution. Schrödinger’s theory seemed (and seems) less abstract. Besides, it provided a clear mathematical description of the wave–particle duality, a concept actively advocated by Einstein. Therefore, many people favored his theory over Heisenberg’s. As we explain in this chapter, both theories are equivalent. However, for many the Schrödinger equation remains the core of quantum mechanics. Erwin Schrödinger had a life with several restless periods in it. He was born and educated in Vienna, where he stayed till 1920, only interrupted by taking up his duty at the Italian front in the first world war. In the years 1920–1921 he successively accepted positions in Jena, Stuttgart, Breslau, and finally Zurich, where he remained for several years. The intense concentration that allowed him to develop his quantum theory he achieved during a Christmas stay in 1925 in a sanatorium in Arosa. But in 1927 he moved again, this time to Berlin. When the Nazis came to power in 1933 he decided to leave the Reich, and so he became itinerant again: Oxford, Edinburgh, Graz, Rome, Ghent ... In 1940 he finally took a position in Dublin where he stayed till his retirement in 1956. In this notation, Hˆ is an operator: it acts on the function ψ and it returns another function, the time derivative of ψ. This operator Hˆ is called the Hamiltonian operator, since it represents the total energy of the particle. This is best seen if we rewrite Hˆ in terms of the momentum operator pˆ ≡ − ih¯∂r and position operator rˆ ≡ r. We then see that Hˆ is identical to (1.3) but with the vectors r and p replaced by their operator analogues. Not only momentum and position, but any physical property can be expressed in terms of operators. For an arbitrary observable A, we can find the corresponding quantum mechanical operator Aˆ. The average value of this observable, sometimes called expectation value, in a given quantum state reads A = dr ψ∗(Aˆψ), (1.6) ψ∗ being the complex conjugate of ψ.
6 Elementary quantum mechanics Control question.Can you write the integral in(1.6)for ()And for (p)? If a state is known tohavea definite value of.let us sayAo.then the wave function of this state must be an eigenfunction of A,i.e.Av=Aov.We indeed see from(1.6)that this would result in (A)=Ao.To prove that not only the expectation value of A is Ao.but also that Ao is actually the only value which can be found for A.we use the definition in(1.6) and compute the fluctuations in the observable a. 的-42=∫rg*ia-([r*aw. (1.7) If in the above formula is an eigenfunction of the operator A.the two terms cancel ons in A ome zer ro.This proves the above statement:any eigenstate of the operator A has a well-defined value for the observable A. As we have already mentioned,the Hamiltonian operator represents the total energy of the particle.Therefore,a state with a definite energy E must obey the eigenvalue equation Hv =Ev. (1.8 We see that the Schrodinger equation predicts a simple time dependence for such a state. (r.t)=exp(-iEt/)(r). (1.9) These states are called stationary states,and the eigenvalue equation (1.8)is called the stationary Schrodinger equation. Control question.Why are these states called stationary.while they retain time dependence?Hint.Consider(1.6). Let us now consider the simple case of a free particle("free"means here"subject to no forces")in an infinite space.This corresponds to setting the potential V(r)to zero or to a constant value everywhere.In this case the solutions of the Schrodinger equation take the form of plane waves p-iE/mep仰·rM. 1 (1.10) where the energy of the particle is purely kinetic.E =p2/2m.The factor 1/results from the normalization condition (1.4).where we assumed that the particle dwells in a large but finite volume V.For a free particle,this is of course artificial.To describe an we have the limity The assumption of a finite mely constructive and we use it throughout the bool We see that the plane wave states(1.10)are eigenfunctions of the momentum oper- ator,pvp =pyp,and therefore all carry a definite momentum p.In a finite volume however.only a discrete subset of all momentum states is allowed.To find this set.we have to take into ac count the boundary onditions at the edges of the volume.Having the An easy and straightforward choice is to take a block volume with dimensionsL Ly xL and assume periodic boundary conditions over the dimensions of the block.This gives the conditions
6 Elementary quantum mechanics Control question. Can you write the integral in (1.6) for x2? And for p2 x ? If a state is known to have a definite value of A, let us say A0, then the wave function of this state must be an eigenfunction of Aˆ, i.e. Aˆψ = A0ψ. We indeed see from (1.6) that this would result in A = A0. To prove that not only the expectation value of A is A0, but also that A0 is actually the only value which can be found for A, we use the definition in (1.6) and compute the fluctuations in the observable A, A2−A 2 = dr ψ∗(AˆAˆψ) − dr ψ∗(Aˆψ) 2 . (1.7) If ψ in the above formula is an eigenfunction of the operator Aˆ, the two terms cancel each other and the fluctuations in A become zero. This proves the above statement: any eigenstate of the operator Aˆ has a well-defined value for the observable A. As we have already mentioned, the Hamiltonian operator represents the total energy of the particle. Therefore, a state with a definite energy E must obey the eigenvalue equation Hˆ ψ = Eψ. (1.8) We see that the Schrödinger equation predicts a simple time dependence for such a state, ψ(r, t) = exp{−iEt/h¯}ψ(r). (1.9) These states are called stationary states, and the eigenvalue equation (1.8) is called the stationary Schrödinger equation. Control question. Why are these states called stationary, while they retain time dependence? Hint. Consider (1.6). Let us now consider the simple case of a free particle (“free” means here “subject to no forces”) in an infinite space. This corresponds to setting the potential V(r) to zero or to a constant value everywhere. In this case the solutions of the Schrödinger equation take the form of plane waves ψp(r, t) = 1 √ V exp{−iEt/h¯} exp{ip · r/h¯}, (1.10) where the energy of the particle is purely kinetic, E = p2/2m. The factor 1/ √ V results from the normalization condition (1.4), where we assumed that the particle dwells in a large but finite volume V. For a free particle, this is of course artificial. To describe an actually infinite space, we have to take the limit V → ∞. The assumption of a finite V, however, is extremely constructive and we use it throughout the book. We see that the plane wave states (1.10) are eigenfunctions of the momentum operator, pˆψp = pψp, and therefore all carry a definite momentum p. In a finite volume, however, only a discrete subset of all momentum states is allowed. To find this set, we have to take into account the boundary conditions at the edges of the volume. Having the volume V is artificial anyway, so we can choose these conditions at will. An easy and straightforward choice is to take a block volume with dimensions Lx ×Ly ×Lz and assume periodic boundary conditions over the dimensions of the block. This gives the conditions
1.3 Diracformulation Vp(x+L.y.z,t)=Vp(x.y+Ly.)=Vp(x.y.+Le.t)Vp(x.y.z.t).resulting in a set of quantized momentum states p=2an(222) 1.11) where n,ny,and n:are integers Control question.Can you derive(1.11)from the periodic boundary conditions and (1.10)yourself? As we see from (1.11),there is a single allowed value of p per volume (h)in momentum space,and therefore the density of momentum states increases with increasing size of the system,D(p)=V/(2h)3.Going toward the limit yoo makes the spac- ing between the discrete momentum values smaller and smaller,and finally results in a continuous spectrum of p. The abov wave e function is one of the simplest solutions of the Schrodinger ation and describesa free particle spread over a volume.nrealhowever functions are usually more complex,and they also can have many components.For exam- ple,if the plane wave in(1.10)describes a single electron,we have to take the spin degree of freedom of the electron into account (see Section 1.7).Since an electron can be in a “spin up”or“spin down”state(or in any superposition of the two).we generally have to use a two-component wave functior (1.12 where the moduli squared of the two components give the relative probabilities to find the electron in the spin up or spin down state as a function of position and time. 1.3 Dirac formulation In the early days of quantum mechanics,the wave function had been thought of as an actual function of space and time coordinates.In this form,it looks very similar to a classical field,i.e.a(multi-component)quantity which is present in every point of coordinate space, such as an electric field E(r,)or a pressure field p(r.).However,it appeared to be very to treat wave functions rathe as elements of a multi-dimensional linear vector space.a Hilbert space.Dirac's formula tion of quantum mechanics enabled a reconciliation of competing approaches to quantum problems and revolutionized the field. In Dirac's approach,every wave function is represented by a vector,which can be put as a"ket"1worba”(1.Ope ators acting on the wave fun tions such as the mentum
7 1.3 Dirac formulation ψp(x + Lx, y,z, t) = ψp(x, y + Ly,z, t) = ψp(x, y,z + Lz, t) = ψp(x, y,z, t), resulting in a set of quantized momentum states p = 2πh¯ nx Lx , ny Ly , nz Lz , (1.11) where nx, ny, and nz are integers. Control question. Can you derive (1.11) from the periodic boundary conditions and (1.10) yourself? As we see from (1.11), there is a single allowed value of p per volume (2πh¯) 3/V in momentum space, and therefore the density of momentum states increases with increasing size of the system, D(p) = V/(2πh¯) 3. Going toward the limit V → ∞ makes the spacing between the discrete momentum values smaller and smaller, and finally results in a continuous spectrum of p. The above plane wave function is one of the simplest solutions of the Schrödinger equation and describes a free particle spread over a volume V. In reality, however, wave functions are usually more complex, and they also can have many components. For example, if the plane wave in (1.10) describes a single electron, we have to take the spin degree of freedom of the electron into account (see Section 1.7). Since an electron can be in a “spin up” or “spin down” state (or in any superposition of the two), we generally have to use a two-component wave function ψp(r, t) = ψp,↑(r, t) ψp,↓(r, t) , (1.12) where the moduli squared of the two components give the relative probabilities to find the electron in the spin up or spin down state as a function of position and time. 1.3 Dirac formulation In the early days of quantum mechanics, the wave function had been thought of as an actual function of space and time coordinates. In this form, it looks very similar to a classical field, i.e. a (multi-component) quantity which is present in every point of coordinate space, such as an electric field E(r, t) or a pressure field p(r, t). However, it appeared to be very restrictive to regard a wave function merely as a function of coordinates, as the Schrödinger formalism implies. In his Ph.D. thesis, Paul Dirac proposed to treat wave functions rather as elements of a multi-dimensional linear vector space, a Hilbert space. Dirac’s formulation of quantum mechanics enabled a reconciliation of competing approaches to quantum problems and revolutionized the field. In Dirac’s approach, every wave function is represented by a vector, which can be put as a “ket” |ψ or “bra” ψ|. Operators acting on the wave functions, such as the momentum
Elementary quantum mechanics and position operator,make a vector out of a vector and can therefore be seen as matrices in this vector space.For instance.a Hamiltonian in general produces )=x) (1.13) An eigenstate of an observable A in this picture can be seen as an eigenvector of the corre sponding matrix.For any real physical quantity,this matrix is Hermitian.that is At=A As we know,an eigenstate of the operator A has a definite value of A,which now simply is the corre onding eige envalue of th trix.By diagonalizing the matrix of an obs 6 one retrieves all possible values which can be f ound when measuring this observable,an the hermiticity of the matrix guarantees that all these eigenvalues are real. One of the definitions of a Hilbert space is that the inner product of two vectors in the space.(vx)=((x))",exists.Let us try to make a connection with the Schrodinger approach.In the Hilbert space of functions of three coordinates r.the Dirac notation implies the pondence)(r)and ((r).and the inner product of tw vectors is defined as (1x)=dr*(r)x(r) 1.14) We see that the normalization condition(1.4)in this notation reads)=1,and the expectation value of an operator A in a state is given by (Al). The dimensionality of the Hilbert space is generally infinite.To give a simple exam- particle apped in a potential well,meaning that →oo when Irl→oo.If one s the Sch finds a set of stationary states,or levels,with a discrete energy spectrum vn(r.t)exp(-iEnt/vn(r). (1.15) where n labels the levels.The wave functions(r)depend on the potential landscape of states of the Hamiltonian can be we denote the basis vectors by In)=n(r,t).we can write any arbitrary vector in this Hilbert space as )=>cnln). (1.16 where the sum in principle runs over all integers. Control question.Can you write the"bra"-version of(1.16)? We note that the basis thus introduced possesses a special handy property:it isr mal,that is,any two basis states satisfy (mln)=8mn. (1.17) Conveniently.the normalized eigenfunctions of any Hermitian operator with non- degenerate(all different)eigenvalues form a proper basis.Making use of orthonormality. we can write any linear operator as A=∑ml (1.18)
8 Elementary quantum mechanics and position operator, make a vector out of a vector and can therefore be seen as matrices in this vector space. For instance, a Hamiltonian Hˆ in general produces Hˆ |ψ=|χ. (1.13) An eigenstate of an observable Aˆ in this picture can be seen as an eigenvector of the corresponding matrix. For any real physical quantity, this matrix is Hermitian, that is Aˆ † = Aˆ. As we know, an eigenstate of the operator Aˆ has a definite value of A, which now simply is the corresponding eigenvalue of the matrix. By diagonalizing the matrix of an observable one retrieves all possible values which can be found when measuring this observable, and the hermiticity of the matrix guarantees that all these eigenvalues are real. One of the definitions of a Hilbert space is that the inner product of two vectors in the space, ψ|χ = (χ|ψ) ∗, exists. Let us try to make a connection with the Schrödinger approach. In the Hilbert space of functions of three coordinates r, the Dirac notation implies the correspondence |ψ↔ψ(r) and ψ|↔ψ∗(r), and the inner product of two vectors is defined as ψ|χ = dr ψ∗(r)χ(r). (1.14) We see that the normalization condition (1.4) in this notation reads | ψ|ψ |2 = 1, and the expectation value of an operator Aˆ in a state ψ is given by ψ|Aˆ|ψ. The dimensionality of the Hilbert space is generally infinite. To give a simple example, let us consider a single spinless particle trapped in a potential well, meaning that V(r) → ∞ when |r|→∞. If one solves the Schrödinger equation for this situation, one finds a set of stationary states, or levels, with a discrete energy spectrum ψn(r, t) = exp{−iEnt/h¯}ψn(r), (1.15) where n labels the levels. The wave functions ψn(r) depend on the potential landscape of the well, and can be quite complicated. This set of eigenstates of the Hamiltonian can be used as a basis of the infinite dimensional Hilbert space. This means that, if we denote the basis vectors by |n ≡ ψn(r, t), we can write any arbitrary vector in this Hilbert space as |ψ = n cn|n, (1.16) where the sum in principle runs over all integers. Control question. Can you write the “bra”-version ψ| of (1.16)? We note that the basis thus introduced possesses a special handy property: it is orthonormal, that is, any two basis states satisfy m|n = δmn. (1.17) Conveniently, the normalized eigenfunctions of any Hermitian operator with nondegenerate (all different) eigenvalues form a proper basis. Making use of orthonormality, we can write any linear operator as Aˆ = nm anm |nm| , (1.18)
1.3 Diracformulation Projections of a vector on the orthogonal axes of a coordinate system.The same point in this two-dimensional space is represented by differntvectors indifferentrdate systems.The representation(inrdinate systemS where the complex numbers anm denote the matrix elements of the operator A,which are defined as anm =(nlAlm). The conve ience of the Dirac formulation is the freedom we have in choosing the basis. As is the case with"real"ve tors,the basis you choose can be seen as a Cartesian coo dinate system,but now in Hilbert space.Let us consider the analogy with usual vectors in two-dimensional space.Any vector then has two components,which are given by the projections of the vector on the orthogonal axes of the coordinate system chosen.When this vector is represented in a different coordinate system,the projections are generally different(see Fig.1.1).The vector,however,is still th me!Smilarly,rbitrary wave function is defined by a set of its are the projections of the wave function on a certain set of basis vectors.In the case of a discrete set of basis states In),an arbitrary quantum state)is written as in(1.16).The projections cn can then be found from cn=川) (1.19 If we were to choose a different set of basis vectors n).the components c we would find would be different,but still represent the same wave function. The same picture holds for systems with a continuous spectrum,since,as explained above,a continuous spectrum can be approximated by a discrete spectrum with infinites- imally small spacing.Wave functions in the Schrodinger equation are written in a coordinate representation.A wave function)can be seen as the projection of the stat v)on the continuous set of basis vectors r).which are the eigenfunctions of the coordi nate operator f.The same wave function can of course also be expressed in the basis of plane waves given in (1.10).In the space spanned by these plane waves Ip),we write w)=∑cplp (1.20) where the components cp are given by Cp =(plv)= (1.21) These components cp can be seen as a different representation of the same wave function. in this case the ntation
9 1.3 Dirac formulation Fig. 1.1 Projections of a vector on the orthogonal axes of a coordinate system. The same point in this two-dimensional space is represented by different vectors in different coordinate systems. The representation (x1,y1) in coordinate system S1 transforms to (x2,y2) in coordinate system S2. where the complex numbers anm denote the matrix elements of the operator Aˆ, which are defined as anm = n|Aˆ|m. The convenience of the Dirac formulation is the freedom we have in choosing the basis. As is the case with “real” vectors, the basis you choose can be seen as a Cartesian coordinate system, but now in Hilbert space. Let us consider the analogy with usual vectors in two-dimensional space. Any vector then has two components, which are given by the projections of the vector on the orthogonal axes of the coordinate system chosen. When this vector is represented in a different coordinate system, the projections are generally different (see Fig. 1.1). The vector, however, is still the same! Similarly, an arbitrary wave function is defined by a set of its components cn, which are the projections of the wave function on a certain set of basis vectors. In the case of a discrete set of basis states |n, an arbitrary quantum state |ψ is written as in (1.16). The projections cn can then be found from cn = n|ψ. (1.19) If we were to choose a different set of basis vectors |n , the components cn we would find would be different, but still represent the same wave function. The same picture holds for systems with a continuous spectrum, since, as explained above, a continuous spectrum can be approximated by a discrete spectrum with infinitesimally small spacing. Wave functions in the Schrödinger equation are written in a coordinate representation. A wave function ψ(r) can be seen as the projection of the state |ψ on the continuous set of basis vectors |r, which are the eigenfunctions of the coordinate operator rˆ. The same wave function can of course also be expressed in the basis of plane waves given in (1.10). In the space spanned by these plane waves |p, we write |ψ = p cp|p, (1.20) where the components cp are given by cp = p|ψ = dr √ V e − i h¯ p·r ψ(r). (1.21) These components cp can be seen as a different representation of the same wave function, in this case the momentum representation.