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Lectures 3-4: Quantum mechanics of one-electron atoms o Schrödinger equation for one-electron atom. The Schrödinger equation o One-electron atom is simplest bound system in nature. o Consists of electron moving in 3D Coulomb potential of nucleus : o Z =1 for hydrogen, Z =2 for helium, etc. o As mass of nucleus (M) >> mass of electron, electron moves moves relative to nucleus as if nucleus was fixed, and the mass (m) of were slightly reduced to µ. o Classically, the total energy of system is: o Solving the Schrödinger equation. o Wavefunctions and eigenvalues. o Atomic orbitals. " = "(x, y,z,t) ! PY3004 PY3004 The Schrödinger equation o The Schrödinger equation Using the Equivalence Principle, the classical dynamical quantities can be replaced with their associated differential operators: o Substituting, we obtain the operator equation: o Assuming electron can be described by a wavefunction of form o Since V(x,y,z) does not depend on time, "(x, y,z,t) = # (x, y,z)e$iEt / h is a solution to the Schrödinger equation. o The time-independent Schrödinger ! equation can therefore be written: o As V = V(r), convenient to use spherical polar coordinates. (1) can write where "2 = or o o This is the Schrodinger equation for the system, where, Laplacian operator. is the PY3004 1 # 2# 1 # # 1 #2 (r )+ 2 (sin$ )+ 2 2 2 r #r #r r sin$ #$ #$ r sin $ #% 2 Can now use separation of variables to split the partial differential equation into a set of ordinary differential equations. ! PY3004 Separation of the Schrödinger equation Separation of the Schrödinger equation o As the LHS of Eqn 3 does nor depend on r or ! and RHS does not depend on " their common value cannot depend on any of these variables. o Assuming that the eigenfunction is separable: o Substituting (2) into the time-independent Schrodinger equation (1) and using the Laplacian: (2) o Therefore set the LHS of Eqn 3 to a constant: (4) and o Carrying out the differentiations, o Note total derivatives now used, as R is a function of r alone, etc. o Now multiply through by "2µr 2 sin 2 # /R$%h 2 and taking transpose, o As LHS only depends on r and RHS on !, both sides must equal a constant, which we choose as l(l+1): (5) (6) o We have now separated the time-independent Schrödinger equation into three differential equations, which each only depend on one of # (4), $ (5) and R(6). (3) ! PY3004 PY3004 Azimuthal solutions (#(")) o A particular solution of (4) is o As the einegfunctions must be single valued, i.e., #(0)=#(2%) => Polar solutions (! (!(!)) o (7) o or using Euler’s formula, 1 = cos ml 2" + isin ml 2" o This is only satisfied if ml = 0, ±1, ±2, ... o Therefore, acceptable solutions to (4) only exist when ml can only have certain integer values, i.e. it is a quantum number. Solutions to Eqn. 7 are of the form where Fl|m | (cos" ) are associated Legendre polynomial functions. ! o By making a change of variables using z = rcos", Eqn. 5 is transformed into a differential equation called the associated Legendre equation: l o ml is called the magnetic quantum number as plays role when atom in magnetic field. PY3004 The requirement that ! remains finite leads to integer values of l and certain ! restrictions on ml as follows: l = 0, 1, 2, 3, ... ml = -l, -l+1, .., 0, .., l-1, l o l is the orbital or angular momentum quantum number. PY3004 Polar solutions (! (!(!)) and spherical harmonics o We can write the associated Legendre functions Spherical harmonic solutions o Y 00= 1 Y 01= cos" Y ±11= (1-cos2")1/2 e±i% Y 02= 1-3cos2" Y ±12= (1-cos2")1/2cos" e±i% with the quantum number subscripts as: # # o !00 = 1 !10 = cos" !20 = 1-3cos2" !2±2 = 1-cos2" The first few spherical harmonics are: !1±1 = (1-cos2")1/2 !2±1 = (1-cos2")1/2cos" Customary to multiply the $(") and !(!) functions to form the so-called spherical harmonic functions which can be written as: i.e., product of trigonometric and polynomial functions. PY3004 PY3004 Radial solutions (R( r )) Radial solutions (R( r )) o What is the ground state of hydrogen (Z=1)? Assuming that the ground state has n = 1, l = 0, the radial wave equation (Eqn. 6) can be written: o In general, the radial wave equation has many solutions, one for each positive integer of n. Can therefore write o Taking the derivative o More generally (Appendix N of Eisberg & Resnick) solutions are of form (7) o Try a solution of the form R = Ae"r / a , where A is a constant and a0 is a constant with the dimension of length. Sub into Eqn. 7: o To satisfy this Eqn. for any r, both expressions in brackets must equal zero. Setting the second expression to zero => where a0 is the Bohr radius. Bound-state solutions are only acceptable if 0 ! o Setting first term to zero => where n is the principle quantum number, defined by n = l +1, l +2, l +3, … eV Same as Bohr’s results PY3004 o En only depends on n: all l states for a given n are degenerate (i.e. have the same energy). PY3004 Radial solutions (R( r )) Summary of separation of variables o Gnl(Zr/a0) are called associated Laguerre polynomials, which depend on n and l. o Express electron wavefunction as product of three functions: " (r,#, $ ) = R(r)%(# )&($ ) o Several resultant radial wavefunctions (Rnl( r )) for the hydrogen atom are given below o As V ! V(t), attempt to solve time-independent Schrodinger equation. o Separate into three ordinary differential equations for R(r),"(# ) and "(# ). o Eqn. 4 for "(# ) only has acceptable solutions for certain value of ml. o Using these values for ml in Eqn. 5, !(!) only has acceptable values for certain ! values of l. o With these values for l in Eqn. 6, R(r) only has acceptable solutions for certain values of En. o => three quantum numbers! ! ! ! PY3004 PY3004 Born interpretation of the wavefunction Born interpretation of the wavefunction o Principle of QM: the wavefunction contains all the dynamical information about the system it describes. o In H-atom, the ground state orbital has the same sign everywhere => sign of orbital must be all positive or all negative. o Born interpretation of the wavefunction: The probability (P(x,t)) of finding a particle at a position between x and x+dx is proportional to |&(x,t)|2dx: o Other orbitals vary in sign. Where orbital changes sign, & = 0 (called a node) => probability of finding electron is zero. o Consider first excited state of hydrogen: sign of wavefunction is insignificant (P = &2 = (-&)2). o P(x,t) is the probability density. o Immediately implies that sign of wavefunction has no direct physical significance. &(x,t) P(x,t) PY3004 PY3004 Born interpretation of the wavefunction o o Radial solutions (R( r )) Next excited state of H-atom is asymmetric about origin. Wavefunction has opposite sign on opposite sides of nucleus. o The radial probability function Pnl(r ), is the probability that the electron is found between r and r + dr: o Some radial probability functions are given at right: o Some points to note: o The r2 factor makes the radial probability density vanish at the origin, even for l = 0 states. The square of the wavefunction is identical on opposite sides, representing equal distribution of electron density on both side of nucleus. o For each state (given n and l), there are n - l - 1 nodes in the distribution. o The distribution for states with l = 0, have n maxima, which increase in amplitude with distance from origin. PY3004 PY3004 Radial solutions (R( r )) Radial probability density o Radial probability distributions for an electron in several of the low energy orbitals of hydrogen. o The abscissa is the radius in units of a0. PY3004 o The radial distribution function, P(r ) gives the probability that the electron will be found in a shell of radius r. o For a 1s electron in hydrogen, P(r ) is a maximum when r = a0/Z. o For hydrogen, Z = 1 => rmax= a0 o Most likely to find the electron at the Bohr radius. PY3004 Hydrogen eigenfunctions s orbitals 0.2 o The eigenfunctions for the state described by the quantum numbers (n, l, ml) are of the form: o Named from “sharp” spectroscopic lines. o l = 0, ml = 0 Depend on quantum numbers: o &n,0,m = Rn,0 (r ) Y0,m (!, ") n = 1, 2, 3, … l = 0, 1, 2, …, n-1 ml = -l, -l+1, …, 0, …, l-1, l o Angular solution: o Value of Y0,0 is constant over sphere. Energy of state on dependent on n: o For n = 0, l = 0, ml = 0 => 1s orbital: o The probability density is 0 -0.2 0.2 0 -0.2 o o En = " o 13.6Z 2 n2 Usually more than one state has same energy, i.e., are degenerate. -0.2 0 0.2 ! PY3004 PY3004 p orbitals d orbitals o Named from “principal” spectroscopic lines. o Named from “diffuse” spectroscopic lines. o l = 1, ml = -1, 0, +1 (n must therefore be >1) o l = 2, ml = -2, -1, 0, +1, +2 (n must therefore be >2) 2 1 o &n,1,m = Rn1 (r ) Y1,m (!, ") o Angular solution: o A node passes through the nucleus and separates the two lobes of each orbital. o Dark/light areas denote opposite sign of the wavefunction. o 0 -1 -2 -1 Three p-orbitals denoted px, py , pz PY3004 o &n,2,m = Rn1 (r ) Y2,m (!, ") o Angular solution: o There are five d-orbitals, denoted o m = 0 is z2. Two orbitals of m = -1 and +1 are xz and yz. Two orbitals with m = -2 and +2 are designated xy and x2-y2. -0.5 0 -1 -0.5 0 0.5 0.5 1 1 PY3004 Quantum numbers o Principal quantum number: o n = 1 (K shell) o n = 2 (L shell) o n = 3 (M shell) o … o Quantum numbers Orbital quantum number: o l = 0 (s subshell) o l = 1 (p subshell) o l = 2 (d subshell) o l = 3 (f subshell) o … o o If n = 1 and l = 0 = > the state is designated 1s. n = 3, l = 2 => 3d state. o The eigenvalues of the one-electron atom depend only on n, but the eigenfunctions depend on n, l and ml, since they are the product of Rnl(r ), $lml (') and !ml(!). o For given n, there are generally several values of l and ml => degenerate eigenfunctions. o Possible values for l and ml for n = 1, 2, 3 n 1 2 3 l 0 0|1 0, 1, 2 ml 0 0 | -1, 0, +1 0 | -1, 0, +1 | -2, -1, 0, 1, 2 Number of degenerate eigenfunctions for each l 1 1|3 1|3|5 Number of degenerate eigenfunctions for each n 1 4 9 See Table 7.1 of Eisberg & Resnick. PY3004 PY3004 Atomic orbitals o Orbital transitions for hydrogen Quantum mechanical equivalent of orbits in Bohr model. o Transition between different energy levels of the hydrogenic atom must follow the following selection rules: (l = ±1 (m = 0, ±1 PY3004 o A Grotrian diagram or a term diagram shows the allowed transitions. o The thicker the line at right, the more probable and hence more intense the transitions. PY3004