Final answer:
The wavefunction for a particle in a one-dimensional infinite potential well (PIB) describes the quantum state of the particle and is given by a sine function multiplied by a normalization constant. The square of the wavefunction's absolute value represents the probability density for finding the particle at a given point within the well.
Step-by-step explanation:
The wavefunction for a particle in a box (PIB) is fundamental in quantum mechanics and describes the state of a quantum particle constrained within a potential well with infinitely high walls, such as an electron confined within a molecule. The wavefunction, often denoted as Ψ(x, t), gives us the probability amplitude of the particle's position and is a key component in determining the quantum behavior of particles. In the case of a one-dimensional particle in an infinite potential well of width L, the normalized wavefunction for the nth quantum state is given by:
Normalized Wavefunction of PIB
Yn(x) = Bn sin(nπx/L)
Here, Bn is the normalization constant, and the sine function reflects the stationary state of the particle with nodes at the boundaries of the box (x=0 and x=L). The quantum number 'n' can be any positive integer and determines the energy level of the particle. Energy levels correspond to different wave functions, and these quantum states are depicted in various quantum mechanical illustrations such as a particle tunneling through a potential barrier.
The square of the absolute value of the wavefunction, |Yn(x)|², represents the probability density of finding the particle at a particular location within the box. Figures depicting bonding and antibonding molecular orbitals for the H2 molecule show how these wavefunctions combine to give an increased or decreased probability density between hydrogen nuclei.
A particle's state of definite energy is known as a stationary state, often visualized by standing wave patterns that do not change over time. Finally, the Born interpretation tells us that the square of the wavefunction gives us the probability density of locating the particle within a certain region of space, which has real physical significance contrary to the wavefunction's amplitude which is just a mathematical construct.