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The Schrödinger equation for the hydrogen atom can be written e2 Ep (1) 2µ 4πε0Υ where e is the charge of an electron (and proton), u is the reduced mass (close to that of an electron) and the Laplacian operator in spherical polar coordinates is 22 1 a2 cos e a r2 sin 0 d0 2 a 1 + + r dr r2 Ə02 p2 sin? 0 dø² If we take a trial wavefunction for the hydrogen 1s orbital to be p x exp(-r/ao) (not nor- malized, but that does not matter here) then we can see that it has no 0 nor o dependence. Using the Laplacian and Eq. 1, get a symbolic formula for ao in terms of the (a) other fundamental constants (ħ, u,e, €o, TI, etc.). Hint. Our trial wavefunction on the right-hand side of Eq. 1 has no (1/r) terms like the left-hand side might.

The Schrödinger equation for the hydrogen atom can be written e2 Ep (1) 2µ 4πε0Υ where e is the charge of an electron (and proton), u is the reduced mass (close to that of an electron) and the Laplacian operator in spherical polar coordinates is 22 1 a2 cos e a r2 sin 0 d0 2 a 1 + + r dr r2 Ə02 p2 sin? 0 dø² If we take a trial wavefunction for the hydrogen 1s orbital to be p x exp(-r/ao) (not nor- malized, but that does not matter here) then we can see that it has no 0 nor o dependence. Using the Laplacian and Eq. 1, get a symbolic formula for ao in terms of the (a) other fundamental constants (ħ, u,e, €o, TI, etc.). Hint. Our trial wavefunction on the right-hand side of Eq. 1 has no (1/r) terms like the left-hand side might.

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The Schrödinger equation for the hydrogen atom can be written e2 2µ ý = Eµ 4περ (1) where e is the charge of an electron (and proton), u is the reduced mass (close to that of an electron) and the Laplacian operator in spherical polar coordinates is 2 a + r dr 1 a2 + + r2 a02 cos e a + r2 sin 0 d0 1 V2 Ər2 p² sin? 0 aq² exp(-r/ao) (not nor- If we take a trial wavefunction for the hydrogen 1s orbital to be y o malized, but that does not matter here) then we can see that it has no 0 nor o dependence. (a) other fundamental constants (ħ, u,e, eo, TT, etc.). Hint. Our trial wavefunction on the right-hand side of Eq. 1 has no (1/r) terms like the left-hand side might. Using the Laplacian and Eq. 1, get a symbolic formula for ao in terms of the Obtain the energy eigenvalue for the hydrogen Is orbital. Express your answer in terms of a0 and other fundamental constants. Hint. Recognize only a single term in the Laplacian does not cancel with the potential energy term in Eq. 1. (b)
Transcribed Image Text:The Schrödinger equation for the hydrogen atom can be written e2 2µ ý = Eµ 4περ (1) where e is the charge of an electron (and proton), u is the reduced mass (close to that of an electron) and the Laplacian operator in spherical polar coordinates is 2 a + r dr 1 a2 + + r2 a02 cos e a + r2 sin 0 d0 1 V2 Ər2 p² sin? 0 aq² exp(-r/ao) (not nor- If we take a trial wavefunction for the hydrogen 1s orbital to be y o malized, but that does not matter here) then we can see that it has no 0 nor o dependence. (a) other fundamental constants (ħ, u,e, eo, TT, etc.). Hint. Our trial wavefunction on the right-hand side of Eq. 1 has no (1/r) terms like the left-hand side might. Using the Laplacian and Eq. 1, get a symbolic formula for ao in terms of the Obtain the energy eigenvalue for the hydrogen Is orbital. Express your answer in terms of a0 and other fundamental constants. Hint. Recognize only a single term in the Laplacian does not cancel with the potential energy term in Eq. 1. (b)
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