Presentation on theme: "3D Schrodinger Equation"— Presentation transcript:
1 3D Schrodinger Equation Simply substitute momentum operatordo particle in box and H atomadded dimensions give more quantum numbers. Can have degeneracies (more than 1 state with same energy). Added complexity.Solve by separating variablesP D S.E.
2 LHS depends on x,y RHS depends on z If V well-behaved can separate further: V(r) or Vx(x)+Vy(y)+Vz(z). Looking at second one:LHS depends on x,y RHS depends on zS = separation constant. Repeat for x and yP D S.E.
3 Example: 2D (~same as 3D) particle in a Square Box solve 2 differential equations and getsymmetry as square. “broken” if rectangleP D S.E.
4 2D gives 2 quantum numbers. Level nx ny Energy 1-1 1 1 2E0 1-2 1 2 5E0 for degenerate levels, wave functions can mix (unless “something” breaks degeneracy: external or internal B/E field, deformation….)this still satisfies S.E. with E=5E0P D S.E.
5 Spherical Coordinates Can solve S.E. if V(r) function only of radial coordinatevolume element issolve by separation of variablesmultiply each side byP D S.E.
6 Spherical Coordinates-Phi Look at phi equation firstconstant (knowing answer allows form)must be single valuedthe theta equation will add a constraint on the m quantum numberP D S.E.
7 Spherical Coordinates-Theta Take phi equation, plug into (theta,r) and rearrangeknowing answer gives form of constant. Gives theta equation which depends on 2 quantum numbers.Associated Legendre equation. Can use either analytical (calculus) or algebraic (group theory) to solve. Do analytical. Start with Legendre equationP D S.E.
8 Spherical Coordinates-Theta Get associated Legendre functions by taking the derivative of the Legendre function. Prove by substitution into Legendre equationNote that power of P determines how many derivatives one can do.Solve Legendre equation by series solutionP D S.E.
9 Solving Legendre Equation Plug series terms into Legendre equationlet k-1=j+2 in first part and k=j in second (think of it as having two independent sums). Combine all terms with same powergives recursion relationshipseries ends if a value equals 0 L=j=integerend up with odd/even (Parity) seriesP D S.E.
10 Solving Legendre Equation Can start making Legendre polynomials. Be in ascending power ordercan now form associated Legendre polynomials. Can only have l derivatives of each Legendre polynomial. Gives constraint on m (theta solution constrains phi solution)P D S.E.
11 Spherical HarmonicsThe product of the theta and phi terms are called Spherical Harmonics. Also occur in E&M.They hold whenever V is function of only r. Seen related to angular momentumP D S.E.
12 3D Schr. Eqn.-Radial Eqn.For V function of radius only. Look at radial equationcan be rewritten as (usually much better...)note L(L+1) term. Angular momentum. Acts like repulsive potential and goes to infinity at r=0 (ala classical mechanics)energy eigenvalues typically depend on 2 quantum numbers (n and L). Only 1/r potentials depend only on n (and true for hydrogen atom only in first order. After adding perturbations due to spin and relativity, depends on n and j=L+s).P D S.E.
13 Particle in spherical box Good first model for nucleiplug into radial equation. Can guess solutionslook first at l=0boundary conditions. R=u/r and must be finite at r=0. Gives B=0. For continuity, must have R=u=0 at r=a. gives sin(ka)=0 andnote plane wave solution. Supplement 8-B discusses scattering, phase shifts. General terms areP D S.E.
14 Particle in spherical box ForLl>0 solutions are Bessel functions. Often arises in scattering off spherically symmetric potentials (like nuclei…..). Can guess shape (also can guess finite well)energy will depend on both quantum numbersand so 1s 1p 1d 2s 2p 2d 3s 3d …………….and ordering (except higher E for higher n,l) depending on detailsgives what nuclei (what Z or N) have filled (sub)shells being different than what atoms have filled electronic shells. In atoms:in nuclei (with j subshells)P D S.E.
15 H Atom Radial Function For V =a/r get (use reduced mass) Laguerre equation. Solutions are Laguerre polynomials. Solve using series solution (after pulling out an exponential factor), get recursion relation, get eigenvalues by having the series end……n is any integer > 0 and L<n. Energy doesn’t depend on L quantum number.Where fine structure constant alpha = 1/137 used. Same as Bohr model energyeigenfunctions depend on both n,L quantum numbers. First few:P D S.E.
17 H Atom DegeneracyAs energy only depends on n, more than one state with same energy for n>1ignore spin for nowEnergy n l m D-13.6 eV (S)-3.4 eV1(P) ,0,-1.5 eV,0,2(D) ,-1,0,1,1 Ground State4 First excited states9 second excited statesP D S.E.
18 Probability Density P is radial probability density small r naturally suppressed by phase space (no volume)can get average, most probable radius, and width (in r) from P(r). (Supplement 8-A)P D S.E.
19 Most probable radius For 1S state Bohr radius (scaled for different levels) is a good approximation of the average or most probable value---depends on n and Lbut electron probability “spread out” with width about the same sizeP D S.E.
21 Radial Probability Density note # nodesP D S.E.
22 Angular Probabilities no phi dependence. If (arbitrarily) have phi be angle around z-axis, this means no x,y dependence to wave function. We’ll see in angular momentum quantizationL=0 states are spherically symmetric. For L>0, individual states are “squished” but in arbitrary direction (unless broken by an external field)Add up probabilities for all m subshells for a given L get a spherically symmetric probability distributionP D S.E.
23 Orthogonality each individual eigenfunction is also orthogonal. Many relationships between spherical harmonics. Important in, e.g., matrix element calculations. Or use raising and lowering operatorsexampleP D S.E.
24 Wave functions build up wavefunctions from eigenfunctions. example what are the expectation values for the energy and the total and z-components of the angular momentum?have wavefunction in eigenfunction componentsP D S.E.