Wave Equation

The wavefunction for a particle is given by the form , where and are real constants. In which one of the following potentials does the particle move?

Following trial wave functions and are used to obtain variational estimates and for the ground state energy of the helium atom. Here, and are variational parameters, and are the position vectors of the electrons. Let be the exact ground state energy of the helium atom. and are the variational estimates of the ground state energy of the helium atom corresponding to and , respectively, Which one of the following options is true?

The wavefunction of a particle in an infinite one-dimensional potential well at time is
where , , and are the normalized ground state, the normalized first excited state, and the normalized second excited state, respectively. , , and are the eigen-energies corresponding to , , and , respectively. The expectation value of energy of the particle in state is