Boundary conditions of the radial Schrodinger equation

In summary, the conversation discusses the radial differential equation obtained by solving the Schrodinger equation using the method of separation of variables. The speaker mentions two sets of boundary conditions, one of which is not actually a boundary condition but rather a means of gaining more information about the solution of the equation. Boundary conditions only come into play after the solution of the equation has been found.
  • #1
spaghetti3451
1,344
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Consider the radial differential equation

##\bigg( - \frac{d^2}{dr^2} + \frac{(l+\frac{d-3}{2})(l+\frac{d-1}{2})}{r^2} + V(r) + m^2 \bigg) \phi_l (r) = \lambda\ \phi_l (r)##,

which I've obtained by solving the Schrodinger equation in ##d## dimensions using the method of separation of variables.

Now, the boundary condition that I have been given is ##\phi_l (r) \sim r^{l+\frac{d-1}{2}}, r \rightarrow 0##.

However, I was expecting the boundary conditions ##\phi_l (0) = 0, \phi^{'}_{l} (0) = 1##.

Does anybody have an idea if there is a relation of some sort between the two sets of boundary conditions in the context of the given differential operator.
 
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  • #2
The first relation isn't a boundary condition! What it means, is that we let r approach zero and see what will be the solution to the limiting equation. Then we demand that the solution to the full equation approaches the solution to the limiting equation. You can do a similar thing for ## r\rightarrow \infty ##. This is called checking the asymptotic behaviour of the equation and is a means of gaining more information about the solution of the equation so that we can solve it easier. Boundary conditions only come into play after we have found the solution of the equation.
 

1. What is the radial Schrodinger equation?

The radial Schrodinger equation is a mathematical equation used in quantum mechanics to describe the behavior of a particle in a central potential field. It takes into account the radial distance of the particle from the center of the potential and allows for the calculation of the particle's energy and wavefunction.

2. What are boundary conditions?

Boundary conditions are conditions that must be satisfied at the boundaries of a system or region. In the context of the radial Schrodinger equation, they refer to the conditions that the wavefunction must meet at the boundaries of the potential field.

3. Why are boundary conditions important in the radial Schrodinger equation?

Boundary conditions are important because they help determine the behavior of the particle in the potential field. They allow us to find the acceptable solutions to the Schrodinger equation and provide information about the energy levels and wavefunctions of the particle.

4. What are the types of boundary conditions in the radial Schrodinger equation?

The types of boundary conditions in the radial Schrodinger equation depend on the type of potential field being studied. For a finite potential well, the boundary conditions are that the wavefunction must approach zero at the boundaries. For an infinite potential well, the wavefunction must be zero at the boundaries. For a harmonic oscillator potential, the wavefunction must be continuous and differentiable at the boundaries.

5. How do boundary conditions affect the solutions to the radial Schrodinger equation?

Boundary conditions play a crucial role in determining the solutions to the radial Schrodinger equation. They limit the possible solutions and help us find the energy levels and wavefunctions of the particle in the potential field. Without satisfying the appropriate boundary conditions, the solutions to the equation would not accurately represent the behavior of the particle.

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