Chapter 8: Spherical Coordinates

Transcription

1 6 Chapte 8: Spheical Coodinates Tiple Integals We've seen that Mathematica can compute integals in Catesian coodinates (x, y, z). Howeve, atoms ae bette descibed using spheical coodinates (, q, f). Hee ae some useful elationships: x = Sin q Cos f y = Sin q Sin f z = Cos j 2 = x 2 + y 2 + z 2 The volume of a 3D object can be thought of as: V =Ÿ Ÿ Ÿ f Hx, y, zl x y z. Using the substitutions above: V = f H Sin q Cos f, Sin q Cos f, Cos ql t whee dt = 2 Sin f q f. The limits of integation ae geneally: 0, 0 q p, 0 f 2p. Fom Physical Chemisty, 6th Edition by Pete Atkins: Example. Nomalize the wavefunction fo the hydogen atom ove all space: y = e - ao, whee ao is the boh adius. Recall that a function is nomalized if A 2 y 2 t =, whee A is the nomalization constant. In spheical coodinates, the integal becomes: Ÿ Ÿ Ÿ A 2 2 y 2 Sin q q f with limits of 0, 0 q p, 0 f 2p. Fo the special case of the H atom, y = R () F (q) G (f), a poduct of functions of each vaiable. Then, the tiple integal can be witten as a poduct of thee sepaate integals: Ÿ A 2 2 R HL 2 Ÿ F HqL 2 Sin q q Ÿ G HfL 2 f In this example, y = y () = R () so F (q) = and G (f) =. Define the wavefunction y and set up the integals one by one. Multiply the integals togethe to get the answe.

2 62 psi = Exp@ ê aod; integal = Integate@A^2 ^2 psi^2, 8, 0, Infinity<D A 2 IfARe@aoD > 0, ao 3, 2 ao 2 E 0 Mathematica sometimes will give a moe complicated answe than what you'e looking fo. The output can be tanslated as: A 2 I ao3 M if ao > ao 2 if ao < 0. Define a new function using the answe above: integalnew = A 2 i ao 3 k A 2 ao 3 y { integalθ =Integate@Sin@θD, 8θ, 0,<D 2 integalφ =Integate@, 8φ, 0,2 <D ans = integalnew integalθ integalφ A 2 ao 3 Solve fo A: Solve@ans, AD êêflatten 9A ao 3ê2 è,a ao 3ê2 è = The nomalization constant A can be chosen to be positive. Define A and pint out the nomalized wavefunction: A = ao 3ê2 è ; Pint@"The nomalized wavefunction is ψ =", A psid

3 63 The nomalized wavefunction is ψ = ao ao 3ê2 è Fom Physical Chemisty, 6th Edition by Pete Atkins: Poblem.8 Nomalize the wavefunction y = x e - / ao to.: Let x = Sin q Cos f so y = Sin q Cos f e - ao. Then y() = e - ao, y(q) = Sin q, and y(f) = Cos f Clea@AD; R = Exp@ ê aod; integal = Integate@A^2 ^2 R^2, 8, 0, Infinity<D A 2 IfARe@aoD > 0, 3ao5 Define a new function fo the output:, 0 2 ao E integalrnew = A 2 i k 3A 2 ao 5 3ao 5 y { F = Sin@θD; integals = Integate@Sin@θD F^2, 8θ, 0,<D 3 G = Cos@φD; integalg = Integate@G^2, 8φ, 0,2 <D ans = integalrnew integals integalg A 2 ao 5

4 6 Solve fo A: AD êêflatten 9A ao 5ê2 è,a ao 5ê2 è = The nomalization constant A can be chosen to be positive. Define A and the nomalized wavefunction y to pint out the answe: A = ao 5ê2 è ; psi = A R F G; Pint@"The nomalized wavefunction is ψ =", psid The nomalized wavefunction is ψ = ao ao 5ê2 è Fom Physical Chemisty, 6th Edition by Pete Atkins: Poblem.9 Nomalize the following wavefunctions to : a) (2 / ao) e b) Sin q Cosf e - / 2ao - / 2ao Fom Physical Chemisty, 6th Edition by Pete Atkins: Poblem.6 Evaluate the expectation values of and 2 fo a hydogen atom with wavefunctions: a) (2 / ao) e b) Sin q Cosf e - / 2ao - / 2ao Recall that the expectation of is: <> = y 2 t and fo <2 > = 2 y 2 t. Use the nomalized wavefunctions fom Poblem.9 to calculate the expectation values.

5 65 Answes: Poblem.9 Pat a: y = (2 / ao) e - / 2ao = y () = R () so F (q) = and G (f) =. Define the wavefunction and integate each integal sepaately: Clea@AD; psi = H2 H ê aoll Exp@ êh2 aold; integal = Integate@A^2 ^2 psi^2, 8, 0, Infinity<D A 2 IfARe@aoD > 0, 8 ao 3, ao 2 I2 0 ao M2 E Define a new function using the answe above: integalnew = A 2 8 ao 3 8A 2 ao 3 integalθ =Integate@Sin@θD, 8θ, 0,<D 2 integalφ =Integate@, 8φ, 0,2 <D ans = integalnew integalθ integalφ 32 A 2 ao 3 Solve fo A: Solve@ans, AD êêflatten 9A ao 3ê2 è,a ao 3ê2 è = The nomalization constant A can be chosen to be positive. Define A and pint out the nomalized wavefunction: A = ao 3ê2 è ; Pint@"The nomalized wavefunction is ψ =", A psid The nomalized wavefunction is ψ = 2ao I2 ao M ao 3ê2 è

6 66 Pat b: y(, q, f) = Sin q Cos f e - / 2ao so R () = e - / 2ao, F (q) = Sin q, and G (f) = Cos f. Clea@AD; R = Exp@ êh2 aold; integalr = Integate@A^2 ^2 R^2, 8, 0, Infinity<D A 2 IfARe@aoD > 0, 2 ao 5, 0 ao E Define a new function fo the output: integalrnew = A 2 2 ao 5 2 A 2 ao 5 F = Sin@θD; integalf = Integate@Sin@θD F^2, 8θ, 0,<D 3 G = Cos@φD; integalg = Integate@G^2, 8φ, 0,2 <D ans = integalrnew integalf integalg 32 A 2 ao 5 Solve fo A: Solve@ans, AD êêflatten 9A ao 5ê2 è,a ao 5ê2 è = The nomalization constant A is positive. Define A and the nomalized wavefunction y to pint out the answe: A = ao 5ê2 è ; psi = A R F G; Pint@"The nomalized wavefunction is ψ =", psid The nomalized wavefunction is ψ = 2ao Cos@φD ao 5ê2 è

7 67 Answes: Poblem.6 Pat a: Fom Poblem.9, the nomalized wavefunction is ψ = Define this as a new function: 2ao I2 ao M ao 3ê2 è. 2ao I2 M ao psi = ao 3ê2 è 2ao I2 ao M ao 3ê2 è The nomalized wavefunction contains only the vaiable and is independent of q and f. This means that you can calculate the expectation of as: Ÿ Ÿ Ÿ y 2 t =Ÿ 3 y 2 Ÿ Sin q q Ÿ f, with limits of 0, 0 q p, 0 q 2p. The expectation of 2 is Ÿ Ÿ Ÿ 2 y 2 t = Ÿ y 2 Ÿ Sin q q Ÿ f. Calculate each individual integal and multiply them togethe to get the answe: integal = Integate@^3 psi^2, 8, 0, Infinity<D IfARe@aoD > 0, 8 ao, 0 ao 3 I2 ao M2 E 32 ao 3 8 ao integalnew = 32 ao 3 3ao integalθ =Integate@Sin@θD, 8θ, 0,<D 2 integalφ =Integate@, 8φ, 0,2 <D ans = integalnew integalθ integalφ; Pint@"<> = ", ansd <> = 6ao

8 68 Fo < 2 >, the only integal that changes in value is the one involving the vaiable. Define a new integal fo and multiply all the integals togethe to get the answe. integal2 = Integate@^ psi^2, 8, 0, Infinity<D IfARe@aoD > 0, 336 ao 5, 0 ao I2 ao M2 E 32 ao 3 integal2new = 2 ao ao5 32 ao 3 ans = integal2new integalθ integalφ; Pint@"< 2 > = ", ansd < 2 > = 2 ao 2 Pat b: Fom Poblem.9, the nomalized wavefunction is ψ = 2ao ao 5ê2 è. This function contains thee vaiables (, q, f), so <> is: Ÿ 3 RHL 2 Ÿ Sin q FHqL 2 q Ÿ GHfL 2 f Define each pat of the wavefunction sepaately and multiply the individual integals togethe to get the answe. R = 2ao ao 5ê2 è 2ao ao 5ê2 è integalr = Integate@^3 R^2, 8, 0, Infinity<D IfARe@aoD > 0, 20 ao 6, 0 ao 5 E 32 ao 5

9 69 integalrnew = 20 ao6 32 ao 5 5 ao F = Sin@θD; integalθ =Integate@Sin@θD F^2, 8θ, 0,<D 3 G = Cos@φD; integalφ =Integate@G^2, 8φ, 0,2 <D ans = integalrnew integalf integalg; Pint@"<> = ", ansd <> = 5ao Fo < 2 >, the only integal that changes in value is the one involving the vaiable. Define a new integal fo and multiply all the integals togethe to get the answe. integal2 = Integate@^ R^2, 8, 0, Infinity<D IfARe@aoD > 0, 720 ao 7, 0 ao 6 E 32 ao 5 integal2new = 720 ao7 32 ao 5 5 ao 2 ans = integal2new integalf integalg; Pint@"< 2 > = ", ansd < 2 > = 30 ao 2