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Chapter 6: Basics of wave mechanics A bit of terminology and formalism The standard basic examples of free and bound states 6.1 Basics and definitions Developed in the 1920’ies by Louis de Broglie, Erwin Schrödinger, Max Born, Werner Heisenberg, Paul Dirac, Wolfgang Pauli and many others. We have to formulate and solve the Schrödinger equation for a given problem. First discuss stationary problems: HfÝrÞ = EfÝrÞ 2 ¥ 42 + E with H = ? 2m pot Ýr Þ the ”Hamilton Operator” (operator of total energy). to solve this Eigenwert problem describe suitable boundary conditions (Randbedingungen) which reduce the infinite amount of solutions of the differential equation to the physically meaningfulwave functions f L ÝrÞ belongingto well defined energy eigenvaluesE L (e.g. n in the Bohr model) with a set of so called quantum numbers L 10.12.01 FU - Physik III - WS 2000/2001 I.V. Hertel 2 Basics and definitions continued. HfÝrÞ = EfÝrÞ with H = 2 2 ¥ ? 2m 4 + E pot Ýr Þ To illustrate the procedure we first treat the 1-D case: 2 d 2 fÝxÞ ¥ ? 2m dx2 10.12.01 + E pot ÝxÞfÝxÞ = EfÝxÞ FU - Physik III - WS 2000/2001 I.V. Hertel 3 6.1.1 Free particle 2 d 2 fÝxÞ ¥ ? 2m dx2 + E pot ÝxÞfÝxÞ = EfÝxÞ ⇒ 2 d 2 fÝxÞ ¥ ? 2m dx2 = EfÝxÞ a trival example: E > 0 , no force acting on the particle (no potential) with k2 = d 2 fÝxÞ dx2 p2 ¥2 = 2mE : 2 ¥ + k 2 fÝxÞ =0 solution fÝxÞ = expÝ+ikxÞ and expÝ?ikxÞ (wave traveling in + and - x direction) wÝxÞ = |fÝxÞ | 2 = 1 is dependentof x! All energies ÝE > 0Þ are physically meaningfull no boundary restrictions: particle can move from x = ?K to +K The most generalsolution is: fÝxÞ =A expÝikxÞ + B expÝikyÞ (does not describe a particle moving freely into one direction ) 10.12.01 FU - Physik III - WS 2000/2001 I.V. Hertel 4 6.1.2 Unbound particle (energy > 0) in a potential In a potential the kinetic energy of the particle changes E kin = E ? E pot ÝxÞ accordingly we rewrite the Schrödinger equation: 2 ¥ 2 d fÝxÞ 2m dx2 d 2 fÝxÞ dx2 + ÝE ? E pot ÝxÞÞfÝxÞ = 0 + k 2 ÝxÞfÝxÞ = 0 with k 2 ÝxÞ = 2mÝE?E pot ÝxÞÞ ¥2 This is not yet a solution of the problem - but a way of writing it a so that we can guess the general behaviour of the fÝxÞ i expÝ?ikÝxÞxÞ solution: Note aside: A realistic, approximativesolution for slowly varying E pot ÝxÞ 2m X ÝE ? E pot ÝxÞ dx is the so called WKB approximation: fÝxÞ i exp ?i ¥ 10.12.01 FU - Physik III - WS 2000/2001 I.V. Hertel 5 6.1.3 Particle in a box (Kastenpotenzial) 2 ¥ 2 d fÝxÞ 2m dx2 + ÝE ? E pot ÝxÞÞfÝxÞ = 0 E pot ÝxÞ = Epot K if x< 0 0 0 ² x< a if K if a² x solution: 0 x 0 fÝxÞ = a x< 0 if expݱikxÞ if 0 ² x< a 0 a² x if generalsolution: fÝxÞ = A expÝ+ikxÞ + B expÝ?ikxÞ ! fÝxÞ = 0 at x = 0 and x = +a boundary conditions: continuity (Stetigkeit): ì standing waves with nodes on boundary fÝxÞ = A sinÝkxÞ ka = n^ (quantum number n = 1, 2, 3. . . . . Þ f n ÝxÞ = A sinÝ n^ a xÞ hence the energies (eigenvalues): E = 10.12.01 ¥ 2k 2 2m = FU - Physik III - WS 2000/2001 I.V. Hertel n 2¥ 2^2 2ma 2 = n2 h2 8ma 2 = n2 E1 6 6.2 General considerations for bound states Bound states: E < 0 basically something like ”standing waves” schrödinger (Albert) E 0 x note: for E ? E pot < 0 (classically forbidden region) k becomes imaginary ì expݱikxÞ í expݱnxÞ with n = 2m ¥ E pot ? E E 0 x we have to fullfill boundary conditions on the left and right: for 2nd order diff. equation that implies continuity and (in general) differentiability (Stetigkeit und Differenzierbarkeit) ì only a set of well defined ”stationary” states with eigenenergies is allowed 10.12.01 FU - Physik III - WS 2000/2001 I.V. Hertel 7 6.3 Observables - 6.3.1 Basic definitions This is an important term in quantum mechanics: Observables are all parameters (variables) which can be measured. e.g. position of a particle x, y, z momentum of a particle p x, p y, p z energy E, angular momentum L 2 projection of L an axis L z etc. Observables are described by operators , say G. There are special states of a quantum system - called eigenstates d of an operator G (also eigenvectors, eigenfunctions), in which this observable assumes well defined values, so called eigenvaluesg. Eigenstates and eigenvaluesof an observableare determinedby solving its eigenvalueequation: Gd L = g L d L Complete set of eigenstates d L ì any state f of the system is a linear superposition: f = > aL d L Orthonormality: Ýd L1 |d L2 Þ = X X X d DL d L2 d 3 r = N L1L 2 1 The most famous exampleof such an eigenvalueequation is the Schrödinger equation Hf = Ef 10.12.01 FU - Physik III - WS 2000/2001 I.V. Hertel 8 6.3.2 Simultaneous measurement of several observables Note: not all observablescan be measured simultaneously or more precisely: .. have simultaneously well defined values e. g. not x and p x (see uncertainty relation AxAp x > h/2) But we may very well measure exactly and simultaneously x and p z etc. Quantum mechanics defines these interrelation in a systematic manner At present we only need to discuss how we describe a measurement Any physical situation can be described by a suitable state (wavefunction), or by an ensembleof states. The probablity to find a particle at r in the volume element d 3 r is wÝrÞd 3 r = |fÝrÞ | 2 d 3 r =|fÝr, S, jÞ | 2 r 2 dr sin S dS dj = |fÝx, y, zÞ | 2 dx dy dz normalisation: X|fÝrÞ | 2 d 3 r = 1 also written: X|fÝrÞ | 2 d 3 r = XfÝrÞf D ÝrÞd 3 r =Ýf|f Þ 10.12.01 FU - Physik III - WS 2000/2001 I.V. Hertel 9 6.3.3 Measurement of an eigenvalue. If of a system is in an Eigenstate d L of the observableG defined by Gd L = g L d L and we make a measurementof G the experimentwill us allways give the value g L Example: The Eigenvalues of the Hamiltonian H of an atomic hydrogen atom are the energies E n = ? E0 2n 2 (as in the Born model). If we know that an H-atom is in the first excited state (quantum number n = 2Þ it is described by an Eigenstate f 2 ÝrÞ and any measurementof the energy of that state will give the identical value E 2 = ? E0 2n 2 But what, if the state under consideration is not an Eigenstate of the observable to be measured? 10.12.01 FU - Physik III - WS 2000/2001 I.V. Hertel 10 6.3.4 Measurement of an expectation value Clearly, we can measure any physically meaningful quantity G for any state f - as often as we want. However, the measurementof observablesG for which the state f under considerationis not an eigenstate(Gf ® gfÞ does not give unique values in the measurements. each measurement can give another value - obeying statistial laws Quantum mechanics makes predictions about the probability to measure a certain observableG: it predicts the average value < G > of the quantity under consideration called expectation value of G ÖGÝrÞ × = X d 3 r f D ÝrÞGÝrÞfÝrÞ 10.12.01 more general: Ö G × = Ýf|Gf Þ FU - Physik III - WS 2000/2001 I.V. Hertel 11 6.3.5 Example and illustration determine the average value of x from a known wavefunction, (in this case the result is trivial x ¯ 0 ) say fÝxÞ = sin x/x Systematically: we divide the x ? scale in boxes of size Ax w(x) count, how often a value of x is measured Ni say: in a total of N measurements we find the particle N 1 times at x1 N 2 times at x2 , N 3 times at x3 etc. Ax x The average value of x is then x x x 1 x = 2 3 1 ÝN x 1 1 N + N 2 x2 + N 3 x3 +. . . Þ = > N i xi N = > wÝxi Þxi N í X|fÝxÞ | 2 xdx expectation value of G : ÖGÝrÞ × = X d 3 r f D ÝrÞGÝrÞfÝrÞ = Ýf |Gf Þ ÖGÝx, y, zÞ × = X X X f D Ýx, y, zÞGÝx, y, zÞfÝx, y, zÞdxdydz in polar coordinates: ÖGÝr, S, jÞ × = X X X f D Ýr, S, jÞGÝr, S, jÞfÝr, S, jÞr 2 dr sin S dS dj for an eigenstated this reduces to: G = Ýd|Gd Þ = Ýd|gd Þ = gÝd|d Þ = g 10.12.01 FU - Physik III - WS 2000/2001 I.V. Hertel 12 6.3.6 Terminology, Orthonormality Summary: Observablesare all parameters which can be measured Eigenstates and eigenvalues of an observable are determinedby solving its eigenvalueequationGd L = g L d L subject to certain boundary conditions the solutions form a complete basis set which usually is orthonormalised: X X X d 3 r d DL1 ÝrÞd L2 ÝrÞ = d L1 |d L2 = N L1L2 Any (pure) state f of the system can be expressed as a linear superposition of the Eigenstates (to any operator) f = > aL d L Eigenstates (and any pure state) are also written as |d L Þ and Ýd L | they are typically characterised by a set of quantum numbers L bra ket hence, simplified , one writes : |LÞ and ÝL| 10.12.01 FU - Physik III - WS 2000/2001 I.V. Hertel 13 6.4 Potential step d 2 fÝxÞ dx2 + 2m ßVÝxÞ ? E àfÝxÞ = 0 2 ¥ E V(x) VÝxÞ = 0 if ?K < x < 0 V0 if 0 ² x< K general solution fÝxÞ =A expÝ+ikxÞ + B expÝ?ikxÞ í +x è ?x k = 2mE ¥ 2mÝE?V 0 Þ ¥ if ?K < x < 0 if 0 ² x< K V0 x Distinguish two cases: 10.12.01 FU - Physik III - WS 2000/2001 I.V. Hertel 14 6.4.1 Potential step 0<E<V0 2mE ¥ k = i fÝxÞ = = real 2mÝV 0 ?EÞ ¥ if = in if ?K < x < 0 V(x) 0 ² x< K V0 E A expÝ+ikxÞ + B expÝ?ikxÞ if ?K < x < 0 C expÝ?nxÞ + D expÝ+nxÞ if x 0 ² x< K no exponential growth and at x = 0 continuity of the function and first derivative: ì B = C?A fÝ0Þ ì A + B = C df dx x=0 ì ikA ? ikB = ?nC 2ik A ? A B = ik?n Ý Þ 10.12.01 ì ikA ? ikÝC ? A Þ = ?nC = 2ik?ik+n A ik?n 2ik A ì C = ik?n Ý Þ = ik+n A ik?n FU - Physik III - WS 2000/2001 I.V. Hertel 15 Potential step 0<E<V0 continued (1) f = A expÝ+ikxÞ + ik+n expÝ?ikxÞ if ?K < x < 0 ik?n 2ik Ýik?n Þ expÝ?nxÞ if 0 ² x< K V(x) V0 E x 10.12.01 FU - Physik III - WS 2000/2001 I.V. Hertel 16 Potential step 0<E<V0 continued (2) f = A expÝ+ikxÞ + ik+n expÝ?ikxÞ if ?K < x < 0 ik?n 2ik Ýik?n Þ expÝ?nxÞ if ψ 2, V(x) V0 0 ² x< K E x the matter wave enters into the classically forbidden region penetration depth 1/n = ¥ 2m ÝV 0 ?E Þ the reflected wave is: B expÝ?ikxÞ = ik+n A expÝ?ikxÞ ik?n 2 f | | We define the current of a travellingwave (or wave packet): j = v|f | 2 = ¥k m In the present case: 2 = ¥k ik+n A B j refl = ¥k | | m ik?n m 10.12.01 2 2 ik+n ?ik+n = ¥k |A | 2 = j = ¥k A | | in m m ik?n ?ik?n FU - Physik III - WS 2000/2001 I.V. Hertel 17