wavefunction

Examples of wavefunction in the following topics:

• The H_2^+ Molecular Ion

  • After the electronic wavefuntion is calculated as a function of $R$, we can determine the proton wavefunction.
  • The proton wavefunction satisfies the one-body Schrodinger equation
  • where $\psi$ is any normalized wavefunction.
  • The $g$ and the $u$ refer to gerade (even-parity) and ungerade (odd-parity) wavefunctions.
  • Because the spatial wavefunction is even with respect interchanging the electrons their spins must be antiparallel.

• Particle in a Box

  • The potential function (V) is time-independent, while the wavefunction itself is time- dependent.
  • The wavefunction must vanish everywhere beyond the edges of the box, as the potential outside of the box is infinite.
  • Furthermore, the amplitude of the wavefunction also may not "jump" abruptly from one point to the next.
  • These two conditions are only satisfied by wavefunctions with the form:
  • Negative values are neglected, since they give wavefunctions identical to the positive solutions except for a physically unimportant sign change.

• Atomic Structure

  • In quantum mechanics we characterize the state of a particles (or group of particles) by the wavefunction ($\Psi$).
  • The wavefunction evolves forward in time according to the time-dependent Schrodinger equation
  • This realization allows us to write the equation that the wavefunction of an atom must satisfy

• B.10 Chapter 10

  • The probability of overlap is simply the squared modulus of the nuclear wavefunction evaluated at $r=0$ integrated over the volume $4/3 \pi (4~\text{fermi})^3$.
  • The nuclear wavefunction is given by
  • This wavefunction is in terms of $r$ as a one-dimensional coordinate; it is analogous to the function $R(r)$ in the expansion of the atomic wavefunction in spherical symmetry.
  • The complete wavefunction is
  • What remains is to evaluate the wavefunctions in both cases, for the electron we have

• A single electron in a central field

  • This means quantum-mechanically that the Hamiltonian commutes with the angular momentum operator, and that the wavefunctions that satisfy the Hamiltonian also are eigenfunction of the angular momentum operator (${\bf L}={\bf r}\times {\bf p}$).We have
  • Because the equation does not depend on $m$, the radial wavefunction only depends on $l$.
  • Because the radial eigenfunctions for different values of $l$ satisfy different equations, there is no orthogonality relation for the radial wavefunctions with different $l$ values.

• Energies of Electron States

  • The most important effect is that when an electron is far from the nucleus the charge of the nucleus is shielded by the other electrons, so wavefunctions that get closer to the nucleus see the full charge of the nucleus and lie lower in energy.
  • we see that the centripetal term is proportional to $l(l+1)$, so we would expect that wavefunctions with larger values of $l$ typically stay further from the nucleus, so we have the rule that for a given value of $n$ states with smaller values of $l$ are more bound.
  • A second important fact is that because electrons are indistinguishable, the wave function of more than one electron must be antisymmetric with respect to interchange of any two electrons (within the axioms of non-relativistic QM it could have be symmetric, but one can prove in relativistic QM that the wavefunction must be antisymmetric—the spin-statistics theorem).
  • Furthermore, we generally try to solve the multielectron problem by assuming that the wavefunction of all the electrons is the antisymmetrized product of single electron wavefunctions,

• Perturbation Theory

  • After we figured out the wavefunctions for the hydrogen atom, we examine the energy states of atoms with more than one electron.
  • Through this process we built up a picture of the structure of atoms from two simple ideas: Schrodinger's equation and that the wavefunction of a bunch of electrons is odd under interchange of any pair of electrons.
  • We remember that the wavefunctions $\psi_f$ are orthonormal so that the integral of a product of two wavefunctions over all space is $\delta_{jf}$.

• Wave Equation for the Hydrogen Atom

  • In addition to mathematical expressions for total angular momentum and angular momentum projection of wavefunctions, an expression for the radial dependence of the wavefunctions must be found.
  • The wavefunction itself is expressed in spherical polar coordinates:

• The Pauli Exclusion Principle

  • The Pauli exclusion principle states that no two fermions can have identical wavefunctions.

• Molecular Excitations

  • where we have solved the angular wavefunction in terms of spherical harmonics like we did for hydrogen.
  • On the other hand because the mass of the ions is much larger than that of the electrons we expect the wavefunction of the ions to be localized in a region $\sim a_0 m/M \ll a_0$.