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The wave function of an initially very localized free particle. In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters ψ and Ψ (lower-case and capital psi, respectively). Wave functions are complex ...
The wave function changes, according to this school of thought, because new information is available. The post-measurement wave function generally cannot be known prior to the measurement, but the probabilities for the different possibilities can be calculated using the Born rule.
= if and only if is exactly equal to the wave function of the ground state of the studied system. The variational principle formulated above is the basis of the variational method used in quantum mechanics and quantum chemistry to find approximations to the ground state.
where is position, is the wave function, is a periodic function with the same periodicity as the crystal, the wave vector is the crystal momentum vector, is Euler's number, and is the imaginary unit. Functions of this form are known as Bloch functions or Bloch states, and serve as a suitable basis for the wave functions or states of electrons ...
A so-called eigenmode is a solution that oscillates in time with a well-defined constant angular frequency ω, so that the temporal part of the wave function takes the form e −iωt = cos(ωt) − i sin(ωt), and the amplitude is a function f(x) of the spatial variable x, giving a separation of variables for the wave function: (,) = ().
The concept of universal wavefunction was introduced by Hugh Everett in his 1956 PhD thesis draft The Theory of the Universal Wave Function. [8] It later received investigation from James Hartle and Stephen Hawking [ 9 ] who derived the Hartle–Hawking solution to the Wheeler–deWitt equation to explain the initial conditions of the Big Bang ...
The variational theorem states that for a time-independent Hamiltonian operator, any trial wave function will have an energy expectation value that is greater than or equal to the true ground-state wave function corresponding to the given Hamiltonian. Because of this, the Hartree–Fock energy is an upper bound to the true ground-state energy ...
The result is that, if energy is measured in units of ħω and distance in units of √ ħ/(mω), then the Hamiltonian simplifies to = +, while the energy eigenfunctions and eigenvalues simplify to Hermite functions and integers offset by a half, = =!