To find the wavefunction of a bound electron in a different energy level, you would typically need to solve the corresponding Schrödinger equation for that particular energy level. However, the specific form of the wavefunction will depend on the potential well or system you are considering.
In the case of the electron in a box, where the electron is confined to a one-dimensional box with infinite potential walls, the energy levels are given by:
E_n = (n^2 * h^2) / (8mL^2),
where n is the quantum number (n = 1, 2, 3, ...), h is Planck's constant, m is the mass of the electron, and L is the length of the box.
For the n=1 energy level, the wavefunction is typically represented by a sinusoidal function within the confines of the box. The specific form depends on the normalization chosen, but a common representation is:
ψ_1(x) = √(2/L) * sin(πx/L),
where x is the position coordinate within the box.
If you want to find the wavefunction for a different energy level, let's say n=2, you would need to solve the Schrödinger equation for that particular energy level, taking into account the corresponding potential well or system. The boundary conditions and the specific form of the potential will determine the mathematical form of the wavefunction for that energy level.
In general, solving the Schrödinger equation for different energy levels involves applying appropriate boundary conditions, solving differential equations, and determining the coefficients or constants in the wavefunction expression. The specific methodology and techniques will depend on the potential well or system under consideration.
It's worth noting that the above discussion assumes a non-relativistic treatment of the electron. For a relativistic treatment, such as in quantum field theory, the wavefunction would be described using a different formalism, such as the Dirac equation.