Finding the Wave Function Eigenfunctions and Energy Eigenvalues in an Infinite Potential Well📂Quantum Mechanics
Finding the Wave Function Eigenfunctions and Energy Eigenvalues in an Infinite Potential Well
Proposition
When the potential takes the form of an infinite well over the interval [0,a], the energy (eigenvalue) En and the wave function (eigenstate) ψn of the wave function are as follows.
A potential of the form shown above U is called an infinite potential well. This system models a situation where a particle cannot leave a certain interval, also known as the particle in a box model. It is a very simple model but serves as an important example where it demonstrates results significantly different from those in classical mechanics. In classical mechanics, the position where a particle may be found is uniform within the interval, whereas in quantum mechanics, the probability of finding the particle varies with its position.
The state with the lowest energy level (=n) among the wave functions is known as the ground state. Any state that is not the ground state is referred to as an excited state.
ground state: first excited state: second excited state: ⋮ψ1(x)=a2sin(aπx)ψ2(x)=a2sin(a2πx)ψ3(x)=a2sin(a3πx)
The wave function cannot have arbitrary energy, but only energies of the form depicted in equation (0). Energy is proportional to n2, and it appears in discrete rather than continuous values. This is called quantization.
Given that the wave function ψ is continuous, the values of the functions obtained in [1] and [2] must match at the boundary. Thus, when x=0, the following must hold:
0=ψ(0)=Asin0+Bcos0=B⟹B=0
The boundary condition must also hold when x=a, therefore:
0=ψ(a)=Asinka(∵B=0)(1)
The sine function is zero for integer n, meaning that solutions exist for all integer n, leading to: