* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download 10.4: Helium Atom - PhysWiki
Scalar field theory wikipedia , lookup
Quantum computing wikipedia , lookup
Renormalization wikipedia , lookup
Quantum field theory wikipedia , lookup
Orchestrated objective reduction wikipedia , lookup
Molecular Hamiltonian wikipedia , lookup
Bohr–Einstein debates wikipedia , lookup
Quantum machine learning wikipedia , lookup
Elementary particle wikipedia , lookup
Measurement in quantum mechanics wikipedia , lookup
Quantum key distribution wikipedia , lookup
Nitrogen-vacancy center wikipedia , lookup
Identical particles wikipedia , lookup
Copenhagen interpretation wikipedia , lookup
Path integral formulation wikipedia , lookup
Quantum group wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Quantum entanglement wikipedia , lookup
History of quantum field theory wikipedia , lookup
Many-worlds interpretation wikipedia , lookup
Quantum teleportation wikipedia , lookup
Wave–particle duality wikipedia , lookup
Perturbation theory (quantum mechanics) wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Spin (physics) wikipedia , lookup
Particle in a box wikipedia , lookup
Atomic orbital wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
Quantum state wikipedia , lookup
Canonical quantization wikipedia , lookup
Electron configuration wikipedia , lookup
Bell's theorem wikipedia , lookup
EPR paradox wikipedia , lookup
Hidden variable theory wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
4/26/2015 10.4: Helium Atom - PhysWiki Sign In Forgot Password Register username username password password Sign In If you like us, please share us on social media. The latest UCD Hyperlibrary newsletter is now complete, check it out. ChemWiki BioWiki GeoWiki StatWiki PhysWiki MathWiki SolarWiki PhysWiki: The Dynamic Physics E-textbook > Quantum Mechanics > Fitzpatrick's Quantum Mechanics > 10: Identical Particles > 10.4: Helium Atom 10.4: Helium Atom Consider the helium atom, which is a good example of a two-electron system. The Hamiltonian is written \[H = \frac {2\,m_1} + \frac{Z\,m_2} - \frac{Z\,e^2}{4\pi\,\epsilon_0\,r_1} -\frac{Z\,e^2}{4\pi\,\epsilon_0\,r_2} + \frac{e^2} {4\pi\,\epsilon_0\,r_{12}},\tag{1081}\] where , , , and . Suppose that the final term on the right-hand side of the above expression were absent. In this case, the overall spatial wavefunction can be formed from products of hydrogen atom wavefunctions calculated with , instead of . Each of these wavefunctions is characterized by the usual triplet of quantum numbers, , , and . Now, the total spin of the system is a constant of the motion (since obviously commutes with the Hamiltonian), so the overall spin state is either the singlet or the triplet state. The corresponding spatial wavefunction is symmetric in the former case, and antisymmetric in the latter. Suppose that one electron has the quantum numbers , , whereas the other has the quantum numbers , , . The corresponding spatial wavefunction is where the plus and minus signs correspond to the singlet and triplet spin states, respectively. Here, wavefunction (calculated with ). For the special case in which the two sets of spatial quantum numbers, is a standard hydrogen atom , , and , , , are the same, the triplet spin state does not exist (because the associated spatial wavefunction is null). Hence, only singlet spin state is allowed, and the spatial wavefunction reduces to (1083) In particular, the ground state ( , , ) can only exist as a singlet spin state (i.e., a state of overall spin 0), and has the spatial wavefunction (1084) where is the Bohr radius. This follows because http://physwiki.ucdavis.edu/Quantum_Mechanics/Fitzpatrick's_Quantum_Mechanics/10%3A_Identical_Particles/10.4%3A_Helium_Atom 1/3 4/26/2015 10.4: Helium Atom - PhysWiki (1085) The energy of this state is (1086) where is the ground state energy of a hydrogen atom. In the above expression, the factor of comes from the fact that there are two electrons in a helium atom. The above estimate for the ground state energy of a helium atom completely ignores the final term on the right-hand side of Equation (1081), which describes the mutual interaction between the two electrons. We can obtain a better estimate for the ground state energy by treating (1084) as the unperturbed wavefunction, and as a perturbation. According to standard first-order perturbation theory, the correction to the ground state energy is (1087) This can be written (1088) since . Now, (1089) where ( ) is the larger (smaller) of and , and is the angle subtended between and . Moreover, the so-called addition theorem for spherical harmonics states that (1090) However, (1091) so we obtain http://physwiki.ucdavis.edu/Quantum_Mechanics/Fitzpatrick's_Quantum_Mechanics/10%3A_Identical_Particles/10.4%3A_Helium_Atom 2/3 4/26/2015 10.4: Helium Atom - PhysWiki (1092) Here, and , and . Thus, our improved estimate for the ground state energy of the helium atom is (1093) This is much closer to the experimental value of than our previous estimate. Consider an excited state of the helium atom in which one electron is in the ground state, while the other is in a state characterized by the quantum numbers , , . We can write the energy of this state as (1094) where is the energy of a hydrogen atom electron whose quantum numbers are the expectation value of . It follows from (1082) (with , , and . According to first-order perturbation theory, is ) that (1095) where (1096) (1097) Here, the plus sign in (1095) corresponds to the spin singlet state, whereas the minus sign corresponds to the spin triplet state. The integral --which is known as the direct integral--is obviously positive. The integral --which is known as the exchange integral--can be shown to also be positive. Hence, we conclude that in excited states of helium the spin singlet state has a higher energy than the spin triplet state. Incidentally, helium in the spin singlet state is known as para-helium, whereas helium in the triplet state is called ortho-helium. As we have seen, for the ground state, only parahelium is possible. The fact that para-helium energy levels lie slightly above corresponding ortho-helium levels is interesting because our original Hamiltonian does not depend on spin. Nevertheless, there is a spin dependent effect--i.e., a helium atom has a lower energy when its electrons possess parallel spins--as a consequence of Fermi-Dirac statistics. To be more exact, the energy is lower in the spin triplet state because the corresponding spatial wavefunction is antisymmetric, causing the electrons to tend to avoid one another (thereby reducing their electrostatic repulsion). Contributors Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin) Powered by MindTouch ® © Copyright 2015 PhysWiki Unless otherwise noted, content in the UC Davis PhysWiki by University of California, Davis is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License. Permissions beyond the scope of this license may be available at [email protected]. http://physwiki.ucdavis.edu/Quantum_Mechanics/Fitzpatrick's_Quantum_Mechanics/10%3A_Identical_Particles/10.4%3A_Helium_Atom 3/3