Question

For a particle confined to a circular area, the simplest quantum mechanics problem you can find can be described by the Schrodinger equation.

−ℏ² /2m ( ∂²u/∂r² +i/r ∂u/∂r )=iℏ ∂u/∂t
with the boundary condition on u(r,t) u(r=R,t)=0
Use the separation of variable to solve it and assume the form u(r,t)=ψ(r)G(t) Here you can assume the function form of G(t) to be like ∼exp(−iEt) and E is the constant in the separation of variable and also eigenvalues. Find all possible eigen values for E. (These are eigenvalues!)

Answer 1

To solve the Schrödinger equation for a particle confined to a circular area, we use the method of separation of variables. Assuming the form u(r,t)=ψ(r)G(t), we separate the equation into two simpler equations for ψ(r) and G(t). By applying the boundary condition u(r=R,t)=0, we can find the possible eigenvalues for E, which represent the energy states of the particle.

Step-by-step explanation:

The Schrödinger equation is a fundamental equation in quantum mechanics that describes the behavior of quantum particles. To solve this equation for a particle confined to a circular area, we use the method of separation of variables. By assuming the wave function can be written as a product of two functions, ψ(r) and G(t), and plugging it into the Schrödinger equation, we can separate the equation into two simpler equations.

For the time-dependent part, G(t), we assume it has the form exp(-iEt), where E is a constant. Substituting this into the equation gives us an equation for ψ(r). The boundary condition u(r=R,t)=0 leads to the requirement that ψ(r) satisfies a certain condition at r=R. Solving this equation gives us the possible eigenvalues for E, which are the allowed energy states of the particle in the circular area.

Therefore, the possible eigenvalues for E are the solutions to the equation obtained from the boundary condition u(r=R,t)=0.