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The normalized 1s wavefunction (n = 1, l = 0) of any one-electron atom or ion can be written as Z3 5 πα 3 where Z is the nuclear charge (number of protons) and a = 100 (r, 0, 0) = R₁0(r)YO (0, ¢) = e -Zr/ao €0h² лm₂е² is the Bohr radius. (a) Construct an integral expression, in r, 0, and that can be used to determine the expectation value for the hydrogen atom potential energy, (V), and evaluate the integrals over and . Be mindful of the integration volume element.

The normalized 1s wavefunction (n = 1, l = 0) of any one-electron atom or ion can be written as Z3 5 πα 3 where Z is the nuclear charge (number of protons) and a = 100 (r, 0, 0) = R₁0(r)YO (0, ¢) = e -Zr/ao €0h² лm₂е² is the Bohr radius. (a) Construct an integral expression, in r, 0, and that can be used to determine the expectation value for the hydrogen atom potential energy, (V), and evaluate the integrals over and . Be mindful of the integration volume element.

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The normalized 1s wavefunction (n = 1, l = 0) of any one-electron atom or ion can be written as 100 (r, 0, 0) = R₁0(r)Y(0,0) = Z3 παρ 3 where Z is the nuclear charge (number of protons) and a = πmpe2 €0h² (a) Construct an integral expression, in r, 0, and that can be used to determine the expectation value for the hydrogen atom potential energy, (V), and evaluate the integrals over and p. Be mindful of the integration volume element. En == e (b) Define x = Zr/ao and use this form to simplify the integral over r and evaluate it (using the tabulated integral provided above) to obtain an expression for (V). Your result should include a factor of V(a). Use the potential energy operator V(r) = where e represents the electron charge and is the permittivity of a vacuum. Express your result in terms of V(a) (that is, express it as something times V(a)). (c) The energy levels of one-electron atoms and ions can be written in terms of a as Z Z²e² (2²72²2) V (ao) = (E) 8περαon2 Using this and your result from part (b), find the expectation value for the kinetic energy, (T), of the electron in 100 (r, 0, $). What is the value of the ratio (T)/(V)? = -Zr/ao Ze² 4π€or is the Bohr radius.
Transcribed Image Text:The normalized 1s wavefunction (n = 1, l = 0) of any one-electron atom or ion can be written as 100 (r, 0, 0) = R₁0(r)Y(0,0) = Z3 παρ 3 where Z is the nuclear charge (number of protons) and a = πmpe2 €0h² (a) Construct an integral expression, in r, 0, and that can be used to determine the expectation value for the hydrogen atom potential energy, (V), and evaluate the integrals over and p. Be mindful of the integration volume element. En == e (b) Define x = Zr/ao and use this form to simplify the integral over r and evaluate it (using the tabulated integral provided above) to obtain an expression for (V). Your result should include a factor of V(a). Use the potential energy operator V(r) = where e represents the electron charge and is the permittivity of a vacuum. Express your result in terms of V(a) (that is, express it as something times V(a)). (c) The energy levels of one-electron atoms and ions can be written in terms of a as Z Z²e² (2²72²2) V (ao) = (E) 8περαon2 Using this and your result from part (b), find the expectation value for the kinetic energy, (T), of the electron in 100 (r, 0, $). What is the value of the ratio (T)/(V)? = -Zr/ao Ze² 4π€or is the Bohr radius.
purpose, the integral S xn e-ßxdr : = will be helpful after you have made a suitable change of coordinate. n! Bn+1
Transcribed Image Text:purpose, the integral S xn e-ßxdr : = will be helpful after you have made a suitable change of coordinate. n! Bn+1
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