For hydrogen-like species, the total energy depends only on the principal quantum number this is not the case for more complex atoms. The ground state is specified by n, the unique set of quantum = 1, = 0, = 0. For the first excited state there are four n l ml independent wavefunctions with quantum numbers: = 2 = 0 = 0 n l ml = 2 = 1 = -1 n l ml = 2 = 1 = 0 n l ml = 2 = 1 = 1 n l ml This means that the first excited state is as the total energy only four-fold degenerate E2 depends on the principle quantum number n. We can define a pseudo-wavefunction which leads to a one dimensional = g(r) r R(r) Schrodinger Equation (10) where the effective potential energy has two contributions due to the Coulomb Ueff interaction and the angular motion of the electron Uc Ul (11) The Matlab mscript qp_pot.m can be used to plot the potential energy functions as shown in figure (4). Fig. 4. Plots of the potential energy functions and for = 0, 1, 2, 3. Ueff, Uc Ul l [qp_pot.m] Probability Distribution Function In the Bohr theory of the hydrogen atom, the electron was pictured as orbiting around the nucleus in a simple circular orbit. The position vector of the electron was well defined. However, in quantum terms the position is not well defined and we must use electron’s
For hydrogen-like species, the total energy depends only on the principal quantum number n, this is not the case for more complex atoms. The ground state is specified by the unique set of quantum n = 1, l = 0, ml = 0. For the first excited state there are four independent wavefunctions with quantum numbers: n = 2 l = 0 ml = 0 n = 2 l = 1 ml = -1 n = 2 l = 1 ml = 0 n = 2 l = 1 ml = 1 This means that the first excited state is four-fold degenerate as the total energy E2 only depends on the principle quantum number n. We can define a pseudo-wavefunction g(r) = r R(r) which leads to a one dimensional Schrodinger Equation (10) where the effective potential energy Ueff has two contributions due to the Coulomb interaction Uc and the angular motion of the electron Ul (11) The Matlab mscript qp_pot.m can be used to plot the potential energy functions as shown in figure (4). Fig. 4. Plots of the potential energy functions Ueff, Uc and Ul for l = 0, 1, 2, 3. [qp_pot.m] Probability Distribution Function In the Bohr theory of the hydrogen atom, the electron was pictured as orbiting around the nucleus in a simple circular orbit. The position vector of the electron was well defined. However, in quantum terms the electron’s position is not well defined and we must use
qp_hydrogen.docx