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Introduction to Differential Equations

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Introduction to Differential Equations Quantum Dynamics – Quick View Concepts of primary interest: The Time-Dependent Schrödinger Equation Probability Density and Mixed States Selection Rules Transition Rates: The Golden Rule Sample Problem Discussions: Tools of the Trade Appendix: Classical E-M Radiation POSSIBLE ADDITIONS: After qualitative section, do the two state system, and then first and second order transitions (follow Fitzpatrick). Chain together the dipole rules to get l = 2,0,-2 and m = -2, -2, … , 2. ??Where do we get magnetic rules? Look at the canonical momentum and the Asquared term. Schrödinger, Erwin (1887-1961) Austrian physicist who invented wave mechanics in 1926. Wave mechanics was a formulation of quantum mechanics independent of Heisenberg's matrix mechanics. Like matrix mechanics, wave mechanics mathematically described the behavior of electrons and atoms. The central equation of wave mechanics is now known as the Schrödinger equation. Solutions to the equation provide probability densities and energy levels of systems. The time-dependent form of the equation describes the dynamics of quantum systems. http://scienceworld.wolfram.com/biography/Schroedinger.html © 1996-2006 Eric W. Weisstein www-history.mcs.st-andrews.ac.uk/Biographies/Schrodinger.html Quantum Dynamics: A Qualitative Introduction Introductory quantum mechanics focuses on time-independent problems leaving Contact: [email protected] dynamics to be discussed in the second term. Energy eigenstates are characterized by probability density distributions that are time-independent (static). There are examples of time-dependent behavior that are by demonstrated by rather simple introductory problems. In the case of a particle in an infinite well with the range [ 0 < x < a], the mixed state below exhibits time-dependence. it (,)xt 1 sinxxeit sin2 e 2 where n2 . a aa n 2ma2 The probability density * for the function (x, t) has the form of a stationary piece plus a piece that oscillates back and forth at the difference frequency 21 = 2 - 1. This oscillation is perhaps the simplest example of quantum dynamics. According to classical E&M, the system radiates light with the oscillation frequency if that oscillating density is a charge density. More is to be said on this topic later. it Exercise: Find the probability density for (,)xt 1 sinxxeit sin2 e 2 a aa assuming that it applies for 0 < x < a and discuss its characteristics. Identify the various time dependences. Classically, Bohr’s orbiting electrons should radiate electromagnetic energy continuously and spiral inward. Bohr postulated that electrons in his special orbits do not radiate, but that they would radiate an electromagnetic energy chunk (a photon) equal to the energy difference between allowed states when the electron in the hydrogen atom made a transition between allowed orbits1. Before launching an attack on quantum dynamics, the origin of classical electromagnetic radiation is to be reviewed. A model for the radiation field can be found in Appendix I. 6/15/2010 Physics Handout Series.Tank: QDyn_QV QMDyn- 2 Classical Electromagnetic Radiation: Charges radiate when they are accelerated. The radiation intensity varies as the square of the sine of the angle between the line of sight direction and that of the acceleration. The radiation electric field is directed oppositely to the component of the acceleration2 that is perpendicular to the line of sight from the field point to the accelerated source charge so an analysis of the acceleration provides information about the direction (polarization) of the electric field in the radiation. The oscillating probability densities for charged particles in mixed states correspond to charge moving and accelerating. Energy eigenstates have static probability density and, should not radiate in this semi-classical view. Be warned: The flow of ideas for describing transitions between quantum states rather than a detailed development is to be presented. Normalizations, relative sizes and numeric factors are omitted. Never use the equations in this handout to calculate a value. That is: This entire handout should be regarded as a Meandering Mind Segment. Stationary and Mixed States: The governing equation for introductory quantum dynamics is the time-dependent Schrödinger equation. 2 i (,) rt Hˆ (,) rt 2 Vr () (,) rt [QMDyn.1] t 2m Consider a state that is an eigenfunction of the Hamiltonian. Hˆ (,)rt E (,) rt i (,) rt nn t n The wavefunction can be separated into temporal and spatial parts: nn(,)rt u () rTn () t 1 Bohr postulated that, in the classical (large radius) limit, the radiated frequency would approach the orbital frequency. This condition is consistent with the one given above, but its nature is not as quantum mechanical. 2 The accerlation is evaluated at the retrarded time, the time at which the radiation was emitted to be observed here now. 6/15/2010 Physics Handout Series.Tank: QDyn_QV QMDyn- 3 leading to the equations Hˆ ur() Eur ()and ETt () i Tt () and hence nnnnnt n itn -1 nnn(,)rt cu () re where n = En and cn is a complex number usually of magnitude 1. For any energy eigenstate, the probability density is stationary (time- independent). 2 itnn it nn(,rt) (,)rt ( u n () re ) u n () re u n () r The probability density is time independent so the particle described by the state is not moving. Exercise: What does it mean to say that a function is an eigenfunction of an operator? What is true about the probability density of an eigenfunction of the hamiltonian? A mixed state includes contributions from two or more energy eigenstates. A simple itn itm mixed state might be of the form mixed (,)rt aurenm () bure () with its associated probability density 2222 it((mn)) * imnt mixed (,)rt a un () r b um () r aburun () m () re ( abun () ru m () re ) The probability density for mixed states has some time-independent components plus components that oscillate at the difference frequencies, |m - n|. An oscillating probability density represents a particle that is accelerating. itn itm Exercise: Show that if mixed (,)rt aurenm () bure () is a mixed state of itn itm uren () and urm ()e which are eigenstates of the full Hamiltonian for the problem, that the probabilities to find the particle with energies corresponding to the states n or m are time-independent. We conclude that the system is not making state- to-state transitions. 6/15/2010 Physics Handout Series.Tank: QDyn_QV QMDyn- 4 That is: this mixed (,)rt does not describe transitions or state evolution. Note that expectation values of various operators in the state mixed can be time dx dependent. For example, /dt may not be zero for the state mixed. itm A general mixed state (,)rt amm u () re does not describe state-to-state m1 transitions. Transitions correspond to the coefficients ak that depend on time. Schrödinger’s equation shows that this is not the case as long as the urm () are eigenstates of the full hamiltonian. State to State Transitions: Mixed states with time independent probability densities are less interesting than the cases in which an electron makes a transition from one quantum state to another. ˆ Consider a quantum problem described by the Hamiltonian H0 that has eigenfunctions itm urem () ˆ itmmit Hu0 mm() re Eum () re . (For definiteness, assume that the states describe the electron in the hydrogen atom.) A general wavefunction for the problem is a mixed state expressed as a sum over the itm eigenfunctions, rt,) cmm u () re . That is: the eigenfunctions form a complete m set and are an orthogonal basis for the space of all physical wavefunctions for the problem. The functions are assumed normalized and orthogonal and hence satisfy the relation: (()ureitkm )* ure ()it dV . all space kkmmmk Begin with the system in a particular pure eigenstate, say the state |n 6/15/2010 Physics Handout Series.Tank: QDyn_QV QMDyn- 5 itn rt,) un () re . The index m is a label representing anyone of the eigenstates. At the initial time, cm = 0 except for cn = 1, or cm(t = to) = mn. At the initial time, there is a 100% probability that a measurement will return a value consistent with the system being in the state n. The time-dependent version of the Schrödinger equation d states that: Hrtˆˆ(,)Hure()itnn Eure () it i . It follows that: 00nnn dt d it it (,rt t ) (,) rt t (,) rt ii Hˆ (,) rtt ure ()nn Eure () t dt 0 nnn. d itit (,)rt tii Hˆ (,) rt t u () renn e E tu () r i tur () dt 0 nn nnn When the system is initially in a pure energy eigenstate n of the full Hamiltonian, time development just adds in more of that same pure state, but ‘- i out of phase’. A system in an eigenstate of the Hamiltonian just remains in that state. Continuously adding a piece ‘-i out of phase’ just changes the complex phase of the function at a rate: d 1 Eu() reitn dt mn . This result follows from the fact that the hamiltonian itn n iure n () operates on an eigenfunction to return a real constant times that same function. This result means that the spatial form un remains fixed, and that every point of the overall form is multiplied by the same complex phase variation, eitn . Exercise: Consider e-it. Show that eeit i() t t e it ()it eit Transitions between eigenstates of the base hamiltonian Ho are caused by interactions which appear as perturbations, new terms added to the Hamiltonian to represent external influences on the quantum system. The basic problem is described by the ˆ itm unperturbed hamiltonian H0 , and its eigen-solutions {… urem () …} provide a complete basis for expanding any well-behaved function defined over the same region ˆ of space. The small interaction term H1 or perturbation is added such that the full 6/15/2010 Physics Handout Series.Tank: QDyn_QV QMDyn- 6 ˆˆ ˆ problem is described by HH0 H1, and the full time development equation is: Hrtˆˆˆ(,) H H (,) rti d 01 dt .
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