Final answer:
The expectation value of the total energy of a particle in a one-dimensional infinite potential well is equivalent to the energy eigenvalue for the nth quantum state, which can be calculated using the quantized energy levels provided by the Schrödinger equation solution for a particle in a box.
Step-by-step explanation:
The question is about finding the expectation value of the total energy of a particle in a one-dimensional infinite potential well, or box, in quantum mechanics. Given that the particle is in a state of definite energy, the potential energy is zero within the box (V(x) = 0 for 0 < x < L).
The expectation value of the total energy in the quantum state can be calculated using the eigenvalues of the energy from the Schrödinger equation.
The energy levels for a particle in a box are quantized and given by the formula Enn=(n^2π^2h^2)/(2mL^2), where n is the principal quantum number, h is Planck's constant, m is the mass of the particle, and L is the width of the box.
For the n-th quantum state, the expectation value for total energy will be exactly the energy eigenvalue Enn, as there is no potential energy contribution within the box.
Since the particle in the infinite potential well cannot have zero energy (due to the Heisenberg uncertainty principle and the nature of quantum systems), the lowest possible energy state (ground state) will have n=1, and this is the minimum energy the particle can possess.