Calculations on the One-dimensional H-Atom in Coordinate and Momentum Space

Transcription

1 Calculations on the One-imensional H-Atom in Coorinate an Momentum Space Frank Riou Chemistry Department CSB SJU The following eercises pertain to several of the l = states of the one-imensional hyrogen atom. s Ψ ( ) ep( ) s Ψ ( ) 8 ( ) ep s Ψ ( ) ep Coorinate Space Calculations Position operator: Momentum operator: p = Integral: i Kinetic energy operator: KE = Potential energy operator: PE = Plot () an () : Ψ ( ) Ψ ( ) Show that this wave function is normalize: Ψ ( ) Ψ ( )

2 Calculate the most probable position for the electron: ( ep( ) ) Calculate the average value of the electron position: = solve Ψ ( ) Ψ ( ) Calculate the probability ensity at both the most probable an average positions of the electron: Most probable: Ψ ( ).54 Average: Ψ.448 Calculate the probability that the electron is between the nucleus an the most probable value of the electron position: Ψ ( ) float. Ψ ( ). Calculate the probability that the electron is between the nucleus an the average value of the electron position: Ψ ( ) float.577 Ψ ( ).577 Calculate the probability that the electron is beyon the most probable position: Ψ ( ) float.677 Ψ ( ).677 Calculate the probability that the electron will be foun insie the nucleus. The nuclear imension is approimately -5 a o.. Ψ ( ).67 4

3 Calculate that position from the nucleus for which the probability of fining the electron is.95: a Given a Ψ ( ) =.95 Fin( a).48 Calculate the uncertainty in position: Δ Ψ ( ) Ψ ( ) Ψ ( ) Ψ ( ) float.866 Calculate the average value of the electron momentum: Ψ ( ) i Ψ ( ) Calculate the uncertainty in momentum: Δp Ψ ( ) Ψ ( ) Ψ ( ) i Ψ ( ) Demonstrate that the position-momentum uncertainty relation is obeye: ΔΔp.866 This value is greater than.5 as require. Calculate the position-momentum commutator. Interpret the result. i Ψ ( ) i Ψ ( ) Ψ ( ) i Position an momentum measurements o not commute. The wave function is not an eigenfunction of the position an momentum operators. Calculate the average value for kinetic energy: Ψ ( ) Ψ ( ) Ψ ( ) Ψ ( ).5

4 Calculate the average value for potential energy: Ψ ( ) Ψ ( ) Ψ ( ) Ψ ( ) Calculate the average value for the total energy:.5 or Ψ ( ) Ψ ( ) Ψ ( ) Ψ ( ) These results illustrate the virial theorem: E = T = V Calculate the probability that the electron is in a classically forbien region, that is a region for which E < V. First show that the classically forbien region begins at =, where E = V. For larger than the potential energy will be greater than the total energy. This is an eample of quantum mechanical tunneling. Show the classically forbien region graphically. = solve Ψ ( ).8.5 Ψ ( ) Demonstrate that the wave function is not an eigenfunction of the kinetic energy operator an comment on the significance of this result: Ψ ( ) Ψ ( )

5 Electron in the hyrogen atom oes not have a well-efine value for kinetic energy. Demonstrate that the wave function is not an eigenfunction of the potential energy operator an comment on the significance of this result: Ψ ( ) Ψ ( ) Electron in the hyrogen atom oes not have a well-efine value for potential energy. Demonstrate that the wave function is an eigenfunction of the total energy operator an comment on the significance of this result: Ψ ( ) Ψ ( ) Ψ ( ) In spite of not having a well-efine kinetic or potential energy, the electron in the hyrogen atom has a well-efine total energy. Ψ ( ) Ψ ( ) = EΨ ( ) solve E What is the energy eigenvalue an how oes it compare to previous calculations in this eercise: The energy eigenvalue is -.5 which is in agreement with <T> + <V> calculate previously. Calculate the overlap integral with the s orbital: Ψ ( ) Ψ ( ) Interpret the result: The orbitals are orthogonal. Calculate the kinetic, potential an total energy of a s electron an show that the virial theorem is satisfie. Ψ ( ) Ψ ( ) 8 Ψ ( ) Ψ ( ) 4 E 8 4 E 8 E = T = V Calculate the kinetic, potential an total energy of a s electron an show that the virial theorem is satisfie. Ψ ( ) Ψ ( ) 8 Ψ ( ) Ψ ( ) 9 E 8 9 E 8 E = T = V

6 Plot the s an s orbitals on the same graph an eplain the orthogonality or net zero overlap. Ψ ( ). From = to the overlap is positive, an = to it is equal in magnitue but negative. Ψ ( ) Ψ ( ) Ψ ( ).88 Ψ ( ) Ψ ( ).88 Momentum Space Calculations Fourier transform the s coorinate-space wave function into momentum space. Φ ( p) π ep( ip) Ψ ( ) π( pi) Plot () an (p) on the same graph: p Ψ ( ) Φ ( p) p Momentum space integral: p Momentum operator: p Kinetic energy operator: p Position operator: i p

7 Demonstrate that the s momentum wavefunction is normalize: Φ ( p) Φ ( p) p or Φ ( p) p Calculate the average value of the momentum an compare it to value obtaine with the coorinate space wave function Φ ( p) p Φ ( p) p Same value Calculate the average value of the kinetic energy an compare it to value obtaine with the s coorinate space wave function p Φ ( p) Φ ( p) p.5 Same value Calculate the average value of the electron position an compare it to value obtaine with the s coorinate space wave function Φ ( p) i p p Φ ( p).5 Same value Calculate the uncertainty in position using the momentum wave function: Φ ( p) p Φ ( p) p Δ Δ.866 Calculate the uncertainty in momentum using the momentum wave function: Φ ( p) p Φ ( p) p Δp Δp Demonstrate that the position-momentum uncertainty relation is satisfie: ΔΔp.866 Same value.

8 Fourier transform the s wavefunction into momentum space: Φ ( p) π ep( ip) Ψ ( ) ( ip) π( ip) Demonstrate the s wavefunction is normalize Φ ( p) p Demonstrate the s an s momentum wavefunctions are orthogonal Φ ( p) Φ ( p) p Fourier transform the s wavefunction into momentum space. Φ ( p) π ep( ip) Ψ ( ) 6( ip) π( ip) 4 Demonstrate the s momentum wavefunction is normalize Φ ( p) p Plot the s, s an s momentum wavefunctions an interpret the graph in terms of the uncertainty principle. Φ ( p) Φ ( p) Φ ( p) p

9 As shown below, as the principal quantum number increases, the spatial istribution of the electron becomes more elocalize. Therefore, accoring to the uncertainty principle, the momentum istribution must become more localize. The graph above shows a more localize momentum istribution as the principle quantum number increases.. Ψ ( ) Ψ ( ) Ψ ( )