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CRC Press eBooks

THE SCHRÖDINGER PICTURE

Mircea S. Rogalski, Stuart B. Palmer

Published Jan 8, 2014• 0 citations
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Abstract

We now may bring the time-dependent state functions ( )tx,Ψ into the picture, taking into account, as we already did in Eq.(29.9), that only the state function but not its position coordinate x varies with time, so that ( ) ( ) t tx dt txd ∂ Ψ∂≡Ψ ,, (30.3) Upon insertion of the expression (29.24) for the expectation value, Eq.(30.2) becomes ˆˆ ˆ ˆ ˆd d fi f i f dx i f f dx i dt dt t t t ⎛ ⎞∂Ψ ∂Ψ ∂= Ψ Ψ = Ψ +Ψ + Ψ Ψ⎜ ⎟∂ ∂ ∂⎝ ⎠∫ ∫ ∫= = = = dx∗ (30.4) On the other hand, as the Hamiltonian Hˆ is a Hermitian operator, we have ( )ˆ ˆˆ ˆ ˆˆ ˆ ˆ, f ff H i Hf f H dx i dt t ∞ ∞ ∂ ∂⎡ ⎤ + = −Ψ Ψ +Ψ Ψ + Ψ Ψ⎣ ⎦ ∂ ∂∫ ∫= = x (30.5) ( ) dx t fidxHffH Ψ∂ ∂Ψ+⎥⎦ ⎤⎢⎣ ⎡ ΨΨ+ΨΨ−= ∫∫ ∗∗ ˆˆˆˆˆ = From Eqs.(30.4) and (30.5) it follows that the evolution equation (30.2) is true only if Ψ=∂ Ψ∂ H t i ˆ= (30.6) and this is a basic result called the time-dependent Schrödinger equation. This equation describes the evolution of quantum systems in terms of time-dependent state functions that obey Eq.(30.6), but neglect the time dependence of the quantum operators. Such a description is known as the Schrödinger picture. The fifth postulate states that the timedependent Schrödinger equation must always be satisfied. Postulate 5. The evolution of the state function of a quantum system is given by the time-dependent Schrödinger equation (30.6), where Hˆ is the Hamiltonian of the system.

DOI
10.1201/b16000-35

Related papers

ATOMIC RADIATION

Mircea S. Rogalski, Stuart B. Palmer

During the presence of the radiation perturbation ( ),ˆ tV the Hamiltonian of the system becomes ( )tVHH ˆˆˆ 0 += (37.4) and, since the state of the system varies with time, we must adopt the time-dependent Schrödinger equation to be a basis for the perturbation theory, that is ( ) ( ) ( )0ˆ ˆti H V tt ∂Ψ ⎡ ⎤= + Ψ⎣ ⎦∂= t (37.5) We assume that Eq.(37.5) has normalized solutions which can be expanded in terms of the as 0kΨ t a t t a t e ε−Ψ = Ψ = Ψ∑ ∑ = (37.6) where the normalization condition on ( )tΨ gives a t =∑ (37.7) that is, ( ) 2ka t gives the probability that the system is in the state at time t. Substituting Eq.(37.6) into Eq.(37.5) yields da t ii a t e V t a t e dt ε εε ε− −⎡ ⎤ ⎡ ⎤− Ψ = +⎢ ⎥ ⎣ ⎦⎣ ⎦∑ ∑ = == = which reduces to da t i e V t a t e dt ε− k ε−Ψ = Ψ∑ ∑= == (37.8) In view of orthogonality of the stationary state eigenfunctions, multiplication by ( ) followed by integration over the configuration space gives da t i e a t e V t dt ε εδ− −= Ψ Ψ∑ ∑= == (37.9) Denoting the matrix elements of the perturbation operator between the stationary states n and k by ( ) ( )0 ˆnk n kV t V t 0= Ψ Ψ (37.10) we can rewrite Eq.(37.9) as da t i a t V t e dt ε ε−= ∑ == (37.11) which is a set of simultaneous differential equations of the first order for the coefficients ( ).na t If we assume that the system is in a definite stationary state at some 0iΨ particular time 0,t = and then it is allowed to interact with the radiation field, we obtain from Eq.(37.6) that ( ) ( )0 00 0i k k aΨ = Ψ = Ψk∑ Multiplying by and integrating gives ( )∗Ψ 0k ( )0ka kiδ= (37.12) that is, and for .( ) 10 =ia ( ) 00 =ka ik ≠ If we consider a time interval t sufficiently short for the coefficients in Eqs.(37.11) to be approximated by the form (37.12), which means that the term gives the greatest contribution to the sum and all other terms can be neglected, we have ( )tak ik = ( ) ( ) ( )0 0 /n ii tn nida ti V t edt ε ε−= == , ( ) ( ) ( )i i iida ti a t Vdt == t (37.13) and on integration we obtain ( ) ( ) ( ) (0 0 / 0 ia t V t e dt n iε ε ′− ′= − ≠∫ == ) , ( ) ( ) tdtVita iii ′′−= ∫ 0 1 = (37.14) The probability (29.29) that the system is in the state at time t, in other words, that coincides with is given by ( )tΨ ,0nΨ ( ) ( ) ( )2 20in n nP t t a t= Ψ Ψ = (37.15) and this is taken to be the probability of direct transition from state to state in a time t, induced by the presence of the radiation field. Assuming a transition where a photon is emitted, that is, we recall the Bohr condition (28.19), which reads ,00 ni εε > (37.16) inni ωεε ==− 00 so that Eqs.(37.14) can be rewritten as ( ) ( ) ( ) 0 n ni ia t V t e dt n iω ′−′ ′= − ≠∫= (37.17) ( ) ( ) 0 1 t i ii ia t V t dt′ ′= − ∫= As we can see from the above, the time-dependent perturbation theory is concerned with transitions that result from perturbation, between the unperturbed levels of the system, while time-independent perturbation theory calculates the stationary shifts in the energy levels. 37.2. FERMI'S GOLDEN RULE Consider the case of a constant perturbation, switched on at time The integrals (37.17) become straightforward, and we obtain .0=t ( ) ( ) ( )1ini tnin in Va t e n iωω −= − ≠= (37.18) ( ) tVita iii =−=1 Substituting Eqs.(37.18) into Eq.(37.6) gives, to a first approximation, the wave function of the atomic system as Vt e t e tωω − − ≠ Ψ = Ψ + − Ψ∑= = (37.19) The probability (37.15) of observing the system in the state at time t, for the coefficients given by Eq.(37.18), reduces to 0 nΨ( )tan ( ) ( 2/sin4 222 2 t V ni in ωω== ) (37.20) which means that transitions from state to states only take place if the matrix elements do not vanish. The plot of niV ( )tPin as a function of inω , given in Figure 37.1 for a perturbation which is on for a time t, suggests that the probability behaves like a Dirac δ -function as t approaches infinity. We may represent the δ -function as ( ) dxe t t 2 1lim 2/ 2/sin lim ωππω ω ( ) ( ininxi dxe in ωδωδπ ω 22/2 1 2/ === ∫ ∞ ∞− ) (37.21) where use was made of a property derived from the normalization condition, written as ( ) ( ) ( ) ( ) ( )u k kukudkuduu δδδδ 1or1 === ∫∫ ∞ P t( ) ω4π6π t2π in t-t-t 2π 4π t 6π t0 Figure 37.1. Dependence of the transition probability on frequency It follows from Figure 37.1 that, in the large-t limit, we have ,0≅inω and hence ( )2sin / 2 / 1.in int tω ω⎡ →⎣ ⎤⎦ Thus, one obtains ( ) ( ) ( ) ( ) (2 24sin / 2 sin / 2 sin / 2lim lim / 2 2/ 2 / 2in in in in int t in inin t t t tt ω ω ω )π πδ ω πδ ωω πωω→∞ →∞ ⎡ ⎤= =⎢ ⎥⎣ ⎦ = so that the transition probability becomes ( ) ( ) (2 2 0 022 2lim in ni in ni i nt t tP t V Vπ π )δ ω δ ε→∞ = = == ε− (37.23) It follows that the transitions take place between states of equal unperturbed energies The transition probability is proportional to the time interval t during which the perturbation is on, that is .00 in εε → ( )tPin approaches infinity as .∞→t Since no experiment lasts for an infinite time, it is the transition probability per unit time, called the transition rate , which is of particular interest. Normally, is part of a continuum, and hence, the transition might end in any state which belongs to a finite range of N final states, rather than in a particular state. We may then define the transition rate as ( )dNtP t P intni ∫∞→→ = 1lim (37.24) The integral over the final states is usually transformed into one over energy, if we define a density of states ( ) ,ρ ε representing the number of continuum states per energy interval, by taking ( )0ndN d 0nρ ε ε= (37.25) so that, substituting Eqs.(37.23) and (37.25), Eq.(37.24) gives ( ) ( ) ( ) ( )20 0 0 0 01 2limi n in n n ni n i ntP P t d Vt πρ ε ε ρ ε δ ε ε→ →∞= =∫ = − (37.26) This result is known as Fermi's golden rule, which states that transitions may occur only between states of equal energies for which the matrix elements of the perturbation operator do not vanish, and that the transition rate is proportional to both the square modulus of these matrix elements and the density of final states. 37.3. EMISSION AND ABSORPTION OF RADIATION Consider the transitions between stationary states which are induced by a harmonic perturbation of the form ( ) ( ) ( ) ( ) ( )( )titi eeArVtArVtV ωωωωω −+== GG ˆcosˆ2ˆ (37.27) It is assumed that the harmonic perturbation corresponds to the interaction of either the electric or magnetic field components of the incoming radiation with the electric or magnetic moment of the atom. Thus, the coupling between the system and the perturbing field can be taken in the form of a field amplitude ( )ωA times a displacement of the system ( ).ˆ rV G On substitution of Eq.(37.27) into Eq.(37.10) one obtains ( ) ( ) ( )( ) ( )( )titinititiinni eeAVeeArVtV ωωωω ωω −− +=+ΨΨ= 00 ˆ G (37.28) It follows from Eq.(37.17) that n ni ia t V A e e dtω ω ω ωω ′ ′− − − +⎡ ⎤ ′= − +⎣ ⎦∫= (37.29) inin eeAV 0 ⎥⎦ ⎤⎢⎣ ⎡ ++−= ωωωω ω ωωωω V A e eω ω ω ωω ω ω ω ω − − − +⎡ ⎤− −= +⎢ ⎥− +⎢ ⎥⎣ ⎦= If the atomic system is bathed into a radiation field of frequency ,inni ωωω −== which implies that according to the Bohr condition (37.16), the transition from state to state describes a process of induced absorption, where the energy of the system is increased by energy transfer from the radiation field to the atom. If 00 in εε > ,ni inω ω ω= = − the second term on the right-hand side of Eq.(37.29) grows without limit, so that the first term can be neglected.

CRC Press eBooks•Jan 8, 2014•0 citations

Temporal Evolution and the Fifth Postulate

M.S. Rogalski, Stuart B. Palmer

Quantum systems, described in terms of wave functions and Hermitian operators that are defined at any instant of time, should be capable of temporal evolution, and this is consistent with the first four Postulates. There is nothing in Postulate 1 to prevent us from considering state vectors Ψ( t ) which depend on time. A Hermitian operator  ( t ), representing a real observable A according to Postulate 2, depends on time either explicitly or through the position coordinate and conjugate momenta, following the correspondence rule introduced by Postulate 4.

CRC Press eBooks•Nov 5, 2020•0 citations

The Time-Dependent Schrödinger Equation

Reinhold A. Bertlmann, Nicolai Friis

Abstract The second chapter introduces the notion of wave functions and their equation of motion proposed by Erwin Schrödinger. The interpretation and normalization of the wave function are discussed, leading to the continuity equation. The chapter also introduces basic mathematical operations that can be carried out for wave functions and operators, describing e.g., how to calculate scalar products and commutators, as well as expectation values and variances of observables. With these tools at hand, Heisenberg’s uncertainty principle is re-examined, focusing on the example of a Gaussian wave packet, which is studied both in position space and momentum space. We also discuss the time evolution of general states and of the exemplary wave packet. Finally, we will see how classical physics is recovered as a limiting case of the quantum-mechanical description

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The Schrödinger Equation

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The postulates of quantum mechanics associate the time-evolution of a system with its time-dependent Schrödinger equation. We start by examining different solutions to this equation for a particle, represented in terms of a Gaussian wave packet, in different scenarios: free, scattered from a potential energy barrier, or trapped in a potential energy well. In some cases, we encounter stationary solutions, in which the probability density does not change in time (a standing wave). These solutions are identified as eigenfunctions of a system Hamiltonian. The properties of the Hamiltonian as a Hermitian operator are introduced, and particularly, the fact that its proper eigenfunctions can compose an orthonormal set, and that the corresponding eigenvalues are real-valued. Learning that all operators that relate to measurables are Hermitian, and that their eigenvalues relate to the measured values, we conclude that the eigenvalues of the energy operator (Hamiltonian) are the energy levels of the quantum system.

Cambridge University Press eBooks•May 11, 2023•0 citations

Light-Matter Interactions

Ying Fu, Min Qiu

We start the discussion by a formal theory of perturbation, which will be further elucidated by straightforward calculations. 2.1.1 The Golden Rule Assume that H0 is a time-independent Hamiltonian and all solutions of the eigenvalue problem H0|k〉 = Ek|k〉 are known; it is straightforward to construct the solution of the equation of motion i ∂|ψ(t)〉 ∂t = H0|ψ(t)〉 (2.1) which coincides at t = 0 with the given initial state |ψ(0)〉. Under these circumstances, the state vector |ψ(t)〉 = e−iH0t/|ψ(0) (2.2) represents the general solution of the equation of motion, Eq. 2.1. Since∑ |k〉〈k| = 1 we obtain |ψ(t)〉 = ∑ k e−iH0t/|k〉〈k|ψ(0)〉 = ∑ k e−iEkt/|k〉〈k|ψ(0)〉 (2.3) We now calculate transition amplitudes between the relevant unperturbed eigenstates due to the appearance of an external perturbation V at t = 0. We transform the equation of motion in the Schro¨dinger picture i ∂|ψ(t)〉 ∂t = (H0 + V )|ψ(t)〉 (2.4) to the interaction picture by a time-dependent unitary operator |ψˆ(t)〉 = exp (iH0t/)|ψ(t)〉 (2.5) so that in the interaction picture, the equation of motion for the state is i ∂|ψˆ(t)〉 ∂t = Vˆ (t)|ψˆ(t)〉 (2.6) where the new interaction operator is given by Vˆ (t) = exp (iH0t/)V exp (−iH0t/) (2.7) Next we introduce a time development operator Tˆ (t) such that |ψˆ(t)〉 = Tˆ (t)|ψˆ(0)〉 (2.8) The time development operator satisfies the integral equation Tˆ (t) = 1− i Vˆ (t)Tˆ (t′)dt′ (2.9) The relationship between the operator expressions in the two pictures is Tˆ (t) = exp (iH0t/)T (t) exp (−iH0t/) (2.10) The transition matrix elements of the time development operator between eigen- states of the unperturbed Hamiltonian are 〈q|Tˆ (t)|k〉 = δ(q − k)− i 〈q|Vˆ (t)Tˆ (t′)|k〉dt′ = δ(q − k)− i 〈q|Vˆ (t′)|k′〉〈k′|Tˆ (t′)|k〉dt′ (2.11) By Eq. 2.10, and note H0|k〉 = Ek|k〉 and H0|q〉 = Eq|q〉, 〈q|Tˆ (t)|k〉 = δ(q − k)− i ei(Eq−Ek′ )t ′/〈q|V (t′)|k′〉〈k′|Tˆ (t′)|k〉dt′ (2.12) The strategy of time-dependent perturbation theory is to assume that V is small and proceed with iterating Eq. 2.9 as a power series in terms of Vˆ Tˆ (t) = 1− i Vˆ ′dt′ − 1 2 Vˆ ′′dt′′ + · · · (2.13) and the transition amplitude from initial state |k〉 to final state |q〉 in the perturbation expansion becomes 〈q|Tˆ (t)|k〉 = δ(q − k)− i − 1 2 × ∫ t′ 0 ei(Ek′−Ek)t ′′/〈k′|V (t′′)|k〉dt′dt′′ + · · · (2.14) It is equivalent to obtain the above equation by iteratively replacing the term 〈k′|Tˆ (t′)|k〉 on the right side of Eq. 2.12 by the whole right-side expression of Eq. 2.12, i.e., ∑ k′ = ∫ t′ 0 + ∑ k′ =k ei(Eq−Ek′ )t ′′/〈q|V (t′′)|k′〉〈k′|Tˆ (t′′)|k〉dt′′ (2.15) By Eq. 2.12 〈k|Tˆ (t′)|k〉 = 1− i ei(Ek−Ek′′ )t ′′/〈k|V (t′′)|k′′〉〈k′′|Tˆ (t′′)|k〉dt′′ (2.16) The first term on the right side of the above equation results in the second term on the right side of Eq. 2.14. For k′ = k and by Eq. 2.12, the second term on the right side of Eq. 2.15 becomes 〈k′|Tˆ (t′)|k〉 = − i ei(Ek′−Ek′′ )t ′′/〈k′|V (t′′)|k′′〉〈k′′|Tˆ (t′′)|k〉dt′′ (2.17) We use Eq. 2.15 to separate the term k′′ = k from k′′ = k in the above equation, which becomes the third term on the right side of Eq. 2.14.

•Aug 8, 2011•0 citations

TIME-DEPENDENT PERTURBATION METHODS

Vishnu S. Mathur, Surendra Singh

We now consider systems for which the Hamiltonian contains time-dependent interaction terms. In such cases we cannot reduce the time-dependent Schro¨dinger equation to an eigenvalue equation. However, if the time-dependent terms can be regarded as small perturbations, we can develop a form of perturbation theory that can be used to calculate the effects of time-dependent terms. Consider a system for which the Hamiltonian may be written as the sum of a dominant unperturbed part Hˆ0 and a small perturbation Hˆ ′ as Hˆ = Hˆ0 + λHˆ ′(t) , (10.1.1) where the unperturbed Hamiltonian Hˆ0 is independent of time, whereas the perturbation Hˆ ′ may depend on time. The parameter 0 < λ < 1 has been introduced to keep track of various orders of approximations as in the case of time-independent perturbation theory discussed in Chapter 8. The basic idea is that a small perturbation to the Hamiltonian will produce a small change in the wave function. For a time-dependent perturbation, this change means that a system, initially in a particular eigenstate of Hˆ0, will find itself in a time-dependent admixture of the eigenstates. Thus the time-dependent perturbation causes the system to undergo transition to different eigenstates of Hˆ0. So for time-dependent perturbations our interest lies in calculating the probabilities for transition to different states.

Chapman & Hall/CRC eBooks•Dec 12, 2008•0 citations
We now may bring the time-dependent state functions ( )tx,Ψ into the picture, taking into account, as we already did in Eq.(29.9), that only the state function but not its position coordinate x varies with time, so that ( ) ( ) t tx dt txd ∂ Ψ∂≡Ψ ,, (30.3) Upon insertion of the expression (29.24) for the expectation value, Eq.(30.2) becomes ˆˆ ˆ ˆ ˆd d fi f i f dx i f f dx i dt dt t t t ⎛ ⎞∂Ψ ∂Ψ ∂= Ψ Ψ = Ψ +Ψ + Ψ Ψ⎜ ⎟∂ ∂ ∂⎝ ⎠∫ ∫ ∫= = = = dx∗ (30.4) On the other hand, as the Hamiltonian Hˆ is a Hermitian operator, we have ( )ˆ ˆˆ ˆ ˆˆ ˆ ˆ, f ff H i Hf f H dx i dt t ∞ ∞ ∂ ∂⎡ ⎤ + = −Ψ Ψ +Ψ Ψ + Ψ Ψ⎣ ⎦ ∂ ∂∫ ∫= = x (30.5) ( ) dx t fidxHffH Ψ∂ ∂Ψ+⎥⎦ ⎤⎢⎣ ⎡ ΨΨ+ΨΨ−= ∫∫ ∗∗ ˆˆˆˆˆ = From Eqs.(30.4) and (30.5) it follows that the evolution equation (30.2) is true only if Ψ=∂ Ψ∂ H t i ˆ= (30.6) and this is a basic result called the time-dependent Schrödinger equation. This equation describes the evolution of quantum systems in terms of time-dependent state functions that obey Eq.(30.6), but neglect the time dependence of the quantum operators. Such a description is known as the Schrödinger picture. The fifth postulate states that the timedependent Schrödinger equation must always be satisfied. Postulate 5. The evolution of the state function of a quantum system is given by the time-dependent Schrödinger equation (30.6), where Hˆ is the Hamiltonian of the system.
During the presence of the radiation perturbation ( ),ˆ tV the Hamiltonian of the system becomes ( )tVHH ˆˆˆ 0 += (37.4) and, since the state of the system varies with time, we must adopt the time-dependent Schrödinger equation to be a basis for the perturbation theory, that is ( ) ( ) ( )0ˆ ˆti H V tt ∂Ψ ⎡ ⎤= + Ψ⎣ ⎦∂= t (37.5) We assume that Eq.(37.5) has normalized solutions which can be expanded in terms of the as 0kΨ t a t t a t e ε−Ψ = Ψ = Ψ∑ ∑ = (37.6) where the normalization condition on ( )tΨ gives a t =∑ (37.7) that is, ( ) 2ka t gives the probability that the system is in the state at time t. Substituting Eq.(37.6) into Eq.(37.5) yields da t ii a t e V t a t e dt ε εε ε− −⎡ ⎤ ⎡ ⎤− Ψ = +⎢ ⎥ ⎣ ⎦⎣ ⎦∑ ∑ = == = which reduces to da t i e V t a t e dt ε− k ε−Ψ = Ψ∑ ∑= == (37.8) In view of orthogonality of the stationary state eigenfunctions, multiplication by ( ) followed by integration over the configuration space gives da t i e a t e V t dt ε εδ− −= Ψ Ψ∑ ∑= == (37.9) Denoting the matrix elements of the perturbation operator between the stationary states n and k by ( ) ( )0 ˆnk n kV t V t 0= Ψ Ψ (37.10) we can rewrite Eq.(37.9) as da t i a t V t e dt ε ε−= ∑ == (37.11) which is a set of simultaneous differential equations of the first order for the coefficients ( ).na t If we assume that the system is in a definite stationary state at some 0iΨ particular time 0,t = and then it is allowed to interact with the radiation field, we obtain from Eq.(37.6) that ( ) ( )0 00 0i k k aΨ = Ψ = Ψk∑ Multiplying by and integrating gives ( )∗Ψ 0k ( )0ka kiδ= (37.12) that is, and for .( ) 10 =ia ( ) 00 =ka ik ≠ If we consider a time interval t sufficiently short for the coefficients in Eqs.(37.11) to be approximated by the form (37.12), which means that the term gives the greatest contribution to the sum and all other terms can be neglected, we have ( )tak ik = ( ) ( ) ( )0 0 /n ii tn nida ti V t edt ε ε−= == , ( ) ( ) ( )i i iida ti a t Vdt == t (37.13) and on integration we obtain ( ) ( ) ( ) (0 0 / 0 ia t V t e dt n iε ε ′− ′= − ≠∫ == ) , ( ) ( ) tdtVita iii ′′−= ∫ 0 1 = (37.14) The probability (29.29) that the system is in the state at time t, in other words, that coincides with is given by ( )tΨ ,0nΨ ( ) ( ) ( )2 20in n nP t t a t= Ψ Ψ = (37.15) and this is taken to be the probability of direct transition from state to state in a time t, induced by the presence of the radiation field. Assuming a transition where a photon is emitted, that is, we recall the Bohr condition (28.19), which reads ,00 ni εε > (37.16) inni ωεε ==− 00 so that Eqs.(37.14) can be rewritten as ( ) ( ) ( ) 0 n ni ia t V t e dt n iω ′−′ ′= − ≠∫= (37.17) ( ) ( ) 0 1 t i ii ia t V t dt′ ′= − ∫= As we can see from the above, the time-dependent perturbation theory is concerned with transitions that result from perturbation, between the unperturbed levels of the system, while time-independent perturbation theory calculates the stationary shifts in the energy levels. 37.2. FERMI'S GOLDEN RULE Consider the case of a constant perturbation, switched on at time The integrals (37.17) become straightforward, and we obtain .0=t ( ) ( ) ( )1ini tnin in Va t e n iωω −= − ≠= (37.18) ( ) tVita iii =−=1 Substituting Eqs.(37.18) into Eq.(37.6) gives, to a first approximation, the wave function of the atomic system as Vt e t e tωω − − ≠ Ψ = Ψ + − Ψ∑= = (37.19) The probability (37.15) of observing the system in the state at time t, for the coefficients given by Eq.(37.18), reduces to 0 nΨ( )tan ( ) ( 2/sin4 222 2 t V ni in ωω== ) (37.20) which means that transitions from state to states only take place if the matrix elements do not vanish. The plot of niV ( )tPin as a function of inω , given in Figure 37.1 for a perturbation which is on for a time t, suggests that the probability behaves like a Dirac δ -function as t approaches infinity. We may represent the δ -function as ( ) dxe t t 2 1lim 2/ 2/sin lim ωππω ω ( ) ( ininxi dxe in ωδωδπ ω 22/2 1 2/ === ∫ ∞ ∞− ) (37.21) where use was made of a property derived from the normalization condition, written as ( ) ( ) ( ) ( ) ( )u k kukudkuduu δδδδ 1or1 === ∫∫ ∞ P t( ) ω4π6π t2π in t-t-t 2π 4π t 6π t0 Figure 37.1. Dependence of the transition probability on frequency It follows from Figure 37.1 that, in the large-t limit, we have ,0≅inω and hence ( )2sin / 2 / 1.in int tω ω⎡ →⎣ ⎤⎦ Thus, one obtains ( ) ( ) ( ) ( ) (2 24sin / 2 sin / 2 sin / 2lim lim / 2 2/ 2 / 2in in in in int t in inin t t t tt ω ω ω )π πδ ω πδ ωω πωω→∞ →∞ ⎡ ⎤= =⎢ ⎥⎣ ⎦ = so that the transition probability becomes ( ) ( ) (2 2 0 022 2lim in ni in ni i nt t tP t V Vπ π )δ ω δ ε→∞ = = == ε− (37.23) It follows that the transitions take place between states of equal unperturbed energies The transition probability is proportional to the time interval t during which the perturbation is on, that is .00 in εε → ( )tPin approaches infinity as .∞→t Since no experiment lasts for an infinite time, it is the transition probability per unit time, called the transition rate , which is of particular interest. Normally, is part of a continuum, and hence, the transition might end in any state which belongs to a finite range of N final states, rather than in a particular state. We may then define the transition rate as ( )dNtP t P intni ∫∞→→ = 1lim (37.24) The integral over the final states is usually transformed into one over energy, if we define a density of states ( ) ,ρ ε representing the number of continuum states per energy interval, by taking ( )0ndN d 0nρ ε ε= (37.25) so that, substituting Eqs.(37.23) and (37.25), Eq.(37.24) gives ( ) ( ) ( ) ( )20 0 0 0 01 2limi n in n n ni n i ntP P t d Vt πρ ε ε ρ ε δ ε ε→ →∞= =∫ = − (37.26) This result is known as Fermi's golden rule, which states that transitions may occur only between states of equal energies for which the matrix elements of the perturbation operator do not vanish, and that the transition rate is proportional to both the square modulus of these matrix elements and the density of final states. 37.3. EMISSION AND ABSORPTION OF RADIATION Consider the transitions between stationary states which are induced by a harmonic perturbation of the form ( ) ( ) ( ) ( ) ( )( )titi eeArVtArVtV ωωωωω −+== GG ˆcosˆ2ˆ (37.27) It is assumed that the harmonic perturbation corresponds to the interaction of either the electric or magnetic field components of the incoming radiation with the electric or magnetic moment of the atom. Thus, the coupling between the system and the perturbing field can be taken in the form of a field amplitude ( )ωA times a displacement of the system ( ).ˆ rV G On substitution of Eq.(37.27) into Eq.(37.10) one obtains ( ) ( ) ( )( ) ( )( )titinititiinni eeAVeeArVtV ωωωω ωω −− +=+ΨΨ= 00 ˆ G (37.28) It follows from Eq.(37.17) that n ni ia t V A e e dtω ω ω ωω ′ ′− − − +⎡ ⎤ ′= − +⎣ ⎦∫= (37.29) inin eeAV 0 ⎥⎦ ⎤⎢⎣ ⎡ ++−= ωωωω ω ωωωω V A e eω ω ω ωω ω ω ω ω − − − +⎡ ⎤− −= +⎢ ⎥− +⎢ ⎥⎣ ⎦= If the atomic system is bathed into a radiation field of frequency ,inni ωωω −== which implies that according to the Bohr condition (37.16), the transition from state to state describes a process of induced absorption, where the energy of the system is increased by energy transfer from the radiation field to the atom. If 00 in εε > ,ni inω ω ω= = − the second term on the right-hand side of Eq.(37.29) grows without limit, so that the first term can be neglected.
Quantum systems, described in terms of wave functions and Hermitian operators that are defined at any instant of time, should be capable of temporal evolution, and this is consistent with the first four Postulates. There is nothing in Postulate 1 to prevent us from considering state vectors Ψ( t ) which depend on time. A Hermitian operator  ( t ), representing a real observable A according to Postulate 2, depends on time either explicitly or through the position coordinate and conjugate momenta, following the correspondence rule introduced by Postulate 4.
Abstract The second chapter introduces the notion of wave functions and their equation of motion proposed by Erwin Schrödinger. The interpretation and normalization of the wave function are discussed, leading to the continuity equation. The chapter also introduces basic mathematical operations that can be carried out for wave functions and operators, describing e.g., how to calculate scalar products and commutators, as well as expectation values and variances of observables. With these tools at hand, Heisenberg’s uncertainty principle is re-examined, focusing on the example of a Gaussian wave packet, which is studied both in position space and momentum space. We also discuss the time evolution of general states and of the exemplary wave packet. Finally, we will see how classical physics is recovered as a limiting case of the quantum-mechanical description
The postulates of quantum mechanics associate the time-evolution of a system with its time-dependent Schrödinger equation. We start by examining different solutions to this equation for a particle, represented in terms of a Gaussian wave packet, in different scenarios: free, scattered from a potential energy barrier, or trapped in a potential energy well. In some cases, we encounter stationary solutions, in which the probability density does not change in time (a standing wave). These solutions are identified as eigenfunctions of a system Hamiltonian. The properties of the Hamiltonian as a Hermitian operator are introduced, and particularly, the fact that its proper eigenfunctions can compose an orthonormal set, and that the corresponding eigenvalues are real-valued. Learning that all operators that relate to measurables are Hermitian, and that their eigenvalues relate to the measured values, we conclude that the eigenvalues of the energy operator (Hamiltonian) are the energy levels of the quantum system.
We start the discussion by a formal theory of perturbation, which will be further elucidated by straightforward calculations. 2.1.1 The Golden Rule Assume that H0 is a time-independent Hamiltonian and all solutions of the eigenvalue problem H0|k〉 = Ek|k〉 are known; it is straightforward to construct the solution of the equation of motion i ∂|ψ(t)〉 ∂t = H0|ψ(t)〉 (2.1) which coincides at t = 0 with the given initial state |ψ(0)〉. Under these circumstances, the state vector |ψ(t)〉 = e−iH0t/|ψ(0) (2.2) represents the general solution of the equation of motion, Eq. 2.1. Since∑ |k〉〈k| = 1 we obtain |ψ(t)〉 = ∑ k e−iH0t/|k〉〈k|ψ(0)〉 = ∑ k e−iEkt/|k〉〈k|ψ(0)〉 (2.3) We now calculate transition amplitudes between the relevant unperturbed eigenstates due to the appearance of an external perturbation V at t = 0. We transform the equation of motion in the Schro¨dinger picture i ∂|ψ(t)〉 ∂t = (H0 + V )|ψ(t)〉 (2.4) to the interaction picture by a time-dependent unitary operator |ψˆ(t)〉 = exp (iH0t/)|ψ(t)〉 (2.5) so that in the interaction picture, the equation of motion for the state is i ∂|ψˆ(t)〉 ∂t = Vˆ (t)|ψˆ(t)〉 (2.6) where the new interaction operator is given by Vˆ (t) = exp (iH0t/)V exp (−iH0t/) (2.7) Next we introduce a time development operator Tˆ (t) such that |ψˆ(t)〉 = Tˆ (t)|ψˆ(0)〉 (2.8) The time development operator satisfies the integral equation Tˆ (t) = 1− i Vˆ (t)Tˆ (t′)dt′ (2.9) The relationship between the operator expressions in the two pictures is Tˆ (t) = exp (iH0t/)T (t) exp (−iH0t/) (2.10) The transition matrix elements of the time development operator between eigen- states of the unperturbed Hamiltonian are 〈q|Tˆ (t)|k〉 = δ(q − k)− i 〈q|Vˆ (t)Tˆ (t′)|k〉dt′ = δ(q − k)− i 〈q|Vˆ (t′)|k′〉〈k′|Tˆ (t′)|k〉dt′ (2.11) By Eq. 2.10, and note H0|k〉 = Ek|k〉 and H0|q〉 = Eq|q〉, 〈q|Tˆ (t)|k〉 = δ(q − k)− i ei(Eq−Ek′ )t ′/〈q|V (t′)|k′〉〈k′|Tˆ (t′)|k〉dt′ (2.12) The strategy of time-dependent perturbation theory is to assume that V is small and proceed with iterating Eq. 2.9 as a power series in terms of Vˆ Tˆ (t) = 1− i Vˆ ′dt′ − 1 2 Vˆ ′′dt′′ + · · · (2.13) and the transition amplitude from initial state |k〉 to final state |q〉 in the perturbation expansion becomes 〈q|Tˆ (t)|k〉 = δ(q − k)− i − 1 2 × ∫ t′ 0 ei(Ek′−Ek)t ′′/〈k′|V (t′′)|k〉dt′dt′′ + · · · (2.14) It is equivalent to obtain the above equation by iteratively replacing the term 〈k′|Tˆ (t′)|k〉 on the right side of Eq. 2.12 by the whole right-side expression of Eq. 2.12, i.e., ∑ k′ = ∫ t′ 0 + ∑ k′ =k ei(Eq−Ek′ )t ′′/〈q|V (t′′)|k′〉〈k′|Tˆ (t′′)|k〉dt′′ (2.15) By Eq. 2.12 〈k|Tˆ (t′)|k〉 = 1− i ei(Ek−Ek′′ )t ′′/〈k|V (t′′)|k′′〉〈k′′|Tˆ (t′′)|k〉dt′′ (2.16) The first term on the right side of the above equation results in the second term on the right side of Eq. 2.14. For k′ = k and by Eq. 2.12, the second term on the right side of Eq. 2.15 becomes 〈k′|Tˆ (t′)|k〉 = − i ei(Ek′−Ek′′ )t ′′/〈k′|V (t′′)|k′′〉〈k′′|Tˆ (t′′)|k〉dt′′ (2.17) We use Eq. 2.15 to separate the term k′′ = k from k′′ = k in the above equation, which becomes the third term on the right side of Eq. 2.14.
We now consider systems for which the Hamiltonian contains time-dependent interaction terms. In such cases we cannot reduce the time-dependent Schro¨dinger equation to an eigenvalue equation. However, if the time-dependent terms can be regarded as small perturbations, we can develop a form of perturbation theory that can be used to calculate the effects of time-dependent terms. Consider a system for which the Hamiltonian may be written as the sum of a dominant unperturbed part Hˆ0 and a small perturbation Hˆ ′ as Hˆ = Hˆ0 + λHˆ ′(t) , (10.1.1) where the unperturbed Hamiltonian Hˆ0 is independent of time, whereas the perturbation Hˆ ′ may depend on time. The parameter 0 < λ < 1 has been introduced to keep track of various orders of approximations as in the case of time-independent perturbation theory discussed in Chapter 8. The basic idea is that a small perturbation to the Hamiltonian will produce a small change in the wave function. For a time-dependent perturbation, this change means that a system, initially in a particular eigenstate of Hˆ0, will find itself in a time-dependent admixture of the eigenstates. Thus the time-dependent perturbation causes the system to undergo transition to different eigenstates of Hˆ0. So for time-dependent perturbations our interest lies in calculating the probabilities for transition to different states.