Question

How to write wave functions for particles?

Answer 1

Any wavefunction that satisfies the schrodinger equation is a possible realisation of a physical system if the function meets some additional criteria, like normalizability and continuity or others, depending on the boundary conditions. So basically you have to solve a partial differential equation.

If you have a particle in a box this means that the potential energy operator in the schrodinger equation has a certain given form. The particular solution for your wavefunction also depends on the initial conditions. Any particular solution can be written as a superposition of basis states (this is possible because the schrodinger equation is linear). These will often be the eigenstates of some self adjoint operator, since those eigenstates form a complete basis of the hilbert space. Often one is looking for the eigenstates of the Hamiltonian. These states are called stationary, because the shape of the probability density of finding the particle somewhere ( the modulus square of the wave function) doesnt change with time for these states.

Finding the respective soulution for any given situation can become very difficult very fast and is achieved by many different methods, often one has to use approximation like the pertubation method (https://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics)) or the the variation method (https://en.wikipedia.org/wiki/Variational_method_(quantum_mechanics)).

Answer 2

$\huge\underline\mathcal{Answer}$

the variational method is one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states. This allows calculating approximate wavefunctions such as molecular orbitals.[1] The basis for this method is the variational principle.[2][3]

The method consists of choosing a "trial wavefunction" depending on one or more parameters, and finding the values of these parameters for which the expectation value of the energy is the lowest possible. The wavefunction obtained by fixing the parameters to such values is then an approximation to the ground state wavefunction, and the expectation value of the energy in that state is an upper bound to the ground state energy. The Hartree–Fock method, Density matrix renormalization group, and Ritz method apply the variational method.

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