First Excited State Energy in a One-Dimensional Potential

Introduction:
In quantum mechanics, the behavior of particles is described by wave functions and energy eigenstates. In a one-dimensional potential, the energy eigenstates represent the allowed energy levels of the particle. The first excited state refers to the second lowest energy level, and the energy associated with this state depends on the specific potential.

One-Dimensional Potential:
The given potential, V(x), is defined as infinite for x less than 0. This means that the particle cannot exist in the region x < 0.="" however,="" for="" x="" greater="" than="" or="" equal="" to="" 0,="" the="" potential="" is="" not="" specified="" and="" can="" take="" any="" />

Quantum Mechanical Solution:
To determine the energy eigenstates and energies for the particle in this potential, we solve the time-independent Schrödinger equation, given by:

Ĥψ(x) = Eψ(x)

Here, Ĥ is the Hamiltonian operator, ψ(x) is the wave function, E is the energy, and x is the position of the particle.

Boundary Conditions:
Since the potential is infinite for x < 0,="" the="" wave="" function="" must="" vanish="" in="" this="" region.="" therefore,="" we="" have="" the="" boundary="" />

ψ(x = 0) = 0

Wave Function and Energy Eigenvalues:
By solving the Schrödinger equation with the given boundary condition, we can find the wave function and corresponding energy eigenvalues for the particle.

For the first excited state, the wave function will have one node. The energy eigenvalue associated with this state can be denoted as E1.

Conclusion:
The first excited state energy of a particle in the given one-dimensional potential is determined by solving the time-independent Schrödinger equation with the appropriate boundary condition. The energy eigenvalue associated with the first excited state is denoted as E1 and depends on the specific potential.