1 Schroenger s Equation for the Hydrogen Atom

Transcription

1 Schroenger s Equation for the Hydrogen Atom Here is the Schroedinger equation in D in spherical polar coordinates. Note that the definitions of θ and φ are the exact reverse of what they are in mathematics. The limits of the three variables are r, θ π, φ π. [ ψ m r + ψ r r + r sin θ ψ ) ] ψ + sin θ θ θ r sin θ φ + Uψ = Eψ ) The potential energy, Ur) = r, depends only on r, so we can write the general solution in separable form, ψr, θ, φ) = Rr)Θθ)Φφ) ) 4πɛ e Inserting into Schroedinger s Equation gives [ ΘΦ d R m dr + r ΘΦdR dr + r sin θ RΦ d Divide by RΘΦ sin θ dθ ) + ] Φ r sin θ RΘd dφ +URΘΦ = ERΘΦ ) m [ d R R dr + dr r R dr + r sin θ Θ d Bring all terms depending on φ to one side sin θ dθ ) + r sin θ Φ d ] Φ dφ + Ur) = E 4) m d [ Φ r sin θ Φ dφ = d R m R dr + dr r R dr + r sin θ Θ d sin θ dθ )] Ur) + E 5) d [ Φ Φ dφ = r sin d R θ R dr + dr r R dr + d r sin θ Θ sin θ dθ )] m r sin θ E Ur)) 6) On the left is a function of only φ, on the right a function of only r and θ. For this to be true for all r, θ, φ, both sides must equal a constant. We still must satisfy boundary conditions: for φ, Φφ + π) = Φφ). Is the constant positive or negative? d Φ Φ dφ = k 7)

2 has solutions Φ = A exp±kφ), and this does not meet the boundary condition. d Φ Φ dφ = +k 8) has solutions Φ = A exp±ikφ), could have been sine and cosine) and this meets the boundary condition providing that expikπ) =, i.e. k = integer negative, positive, zero). The details of the remainder of the solution will be left for other courses. We will just use the results. Quantum numbers for the Hydrogen Atom The full solution requires quantum numbers, and the boundary conditions place specific requirements on these. Principle quantum number determines energy) n =,,,... Angular momentum quantum number orbital qn) l =,,..., n- Magnetic quantum number m l = l, l +,..., l, l Here are possible sets of quantum numbers: n l m l Also called s s -,, p s -,, p -, -,,, d The common names s, p, d, etc.) give the principle quantum number, and then use a letter to indicate the angular momentum quantum number. The letters date to the days of spectroscopic identification, s for sharp lines, p for principal lines intense), d for diffuse lines, f l = ) for finely spaced lines. After f the letters go alphabetically, g, h, i, j. Wavefunctions for Hydrogen Earlier we had ψ = Rr)Θθ)Φφ). The boundary conditions that lead to the three quantum numbers also are used to describe the three functions in ψ. Each part is separately normalized.

3 . Volume element in spherical polar coordinates In Cartesian coordinates, an infinitesimal volume element is dv = dx dy dz. For spherical polar coordinates this becomes dv = r sin θ dr dφ 9) The volume of a sphere of radius R is then V = R π π r dr sin θ dφ = R ) )π) = 4 πr ). Φ depends on m l Our solution was Doing the φ normalization integral, Φ = A expi m l φ) ) π π Φ Φdφ = A dφ = A π) = A = π ) Hence Φφ) = π expim l φ) ). Θ depends on l, m l The normalization integral for Θ is π Some of the normalized solutions are Θ Θ sinθ) = 4)

4 l m l Θ l,ml θ) l m l Θ l,ml θ) 4 cos θ ) ± cos θ ± sin θ ± 5 sin θ cos θ 5 4 sin θ.4 Rr) depends on n, l The normalization integral is Here are the first few radial wavefunctions. r R R dr = 5) n l R n,l r) n l R n,l r) a ) / a / ) exp ra ra ) exp r ) a r exp r ) a ) / a a 8 a / a / 7 8 ra + r 4 8 a / a ) exp r ) a 6 r ) r exp r ) a a a r a exp r ) a 4 Energy levels for Hydrogen Solving Schroedinger s equation gives us the energy, and it exactly the same as the expression for energy in Bohr s model, Z is charge of nucleus, µ is the reduced mass of the system) E n = Z µe 4 π ɛ n =.6eV n for hydrogen 6) 4

5 Notice that l and m l have no effect on the energy. Thus the energy levels are degenerate. Often the energy level diagram is drawn as in Figure 7. in the text, or something similar. We will shortly discuss the spin quantum number, and it will double the degeneracies. n l State Degeneracy Degeneracy with Spin s, s, p 4 8,, s, p, d 9 8 4,,, 4s, 4p, 4d, 4f 6 If we look at the periodic chart of the elements we see elements H, He) in the first row, 8 elements Li to Ne) in the second row, 8 elements K to Kr) in the fourth row, and elements Cs to Rn including the Lanthanide series) in the sixth row. These match some of the numbers in the above table, giving us confidence that we may be on the right track for more than just hydrogen! 5 Using Radial Probability Density Function The radial probability function, i.e. the probability, dp of finding the electron in a region between r and r + dr is dp r) = r R dr 7) There are three types of questions that we can ask using the radial probability density: What is the most likely location in which to find an electron? What is the probability of finding the electron in some range r r r? What it the average value of some function of r? The radial probability density function is plotted in Figure 7.4 in the text. The most likely position to find an electron is where the plot reaches a maximum. You should be able to do this using calculus. To find the probability of finding the electron in some range, do the integral probability = r r r R dr 8) This assumes that the potential energy is just the Coulomb potential energy. Careful measurements show that there are other small contributions to the potential energy arising from magnetic effects of the nucleus on the electron, for example. 5

6 To find the average of a function, for all radii, 6 Angular Momentum < fr) >= fr) r R dr 9) The quantum number l is used to get the magnitude of the angular momentum. L = ll + ) ) The quantum number m l allows us to write the value of one component the z-component since that is where the polar angle is measured from) L z = m l ) An uncertainty principle holds for the components of angular momentum. What we do know is L x + L y = L L z ) but we do not know individually what L x or L y are. To get individual components in x or y, we would need to know φ: this can be written as the uncertainty principle L z φ ) We can know E, L, L z and θ simultaneously to perfect precision. cos θ = L z L = m l ll + ) 4) Notice the possible values of the angular momentum are in the range pay attention to the <, it means less than, not less than or equal to) l < L < n 5) 6