Simple question on Schrodinger Eqn

In summary, the substitution of \sigma (r) = r X(x) in the Schrödinger Equation for spherically symmetric systems simplifies the equation and allows for a relationship with the one-dimensional case. The boundary condition of a finite wave function at all points is necessary for physical reasons, and the wave function must also be continuous and differentiable at all points. The symmetry of the wave function depends on the potential being used.
  • #1
mar0
2
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I take Schrödinger Equation and look for spherically symmetric stationary state wave function. I make substitution [tex]\sigma (r) = r X(x)[/tex] get simpler form. This equation is defined for [tex] r \geq 0 [/tex] , Boundary Conditions is that wavefunction is finite at all points. Now It is possible to relate nicely with one dimensional case by extending potential to be even and define [tex] r [/tex] on the whole real line.

My question is: Probably I am lacking mathematical insight, but why such substitution relates so perfectly well? Then further boundary condition is that [tex]\sigma (r)[/tex] is normalizable, or actually what we really need is square integrable for [tex] r \geq 0 [/tex]. What are further constraints for this "wave function"? It must be either odd or even. My textbook says it must be odd, so why is that?

Original wave function will be finite at origin provided [tex]\sigma (r)[/tex] is zero there.

Many thanks.

///Never mind (probably this was supposed to be in homework section anyway)
 
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  • #2

Thank you for your post. The substitution \sigma (r) = r X(x) is commonly used in solving the Schrödinger Equation for spherically symmetric systems because it simplifies the equation and allows us to relate it to the one-dimensional case. This substitution essentially separates the radial and angular components of the wave function, making it easier to solve.

The boundary condition that the wave function is finite at all points is necessary for physical reasons. If the wave function is not finite, it means that the probability of finding the particle at that point is infinite, which is not physically meaningful.

As for the further constraints on the wave function, it must also be continuous and differentiable at all points, including the boundary. This is necessary for the wave function to be well-behaved and physically meaningful.

As for the odd or even symmetry of the wave function, it depends on the specific potential being used. In some cases, the potential may be even and thus the wave function must also be even in order to satisfy the boundary condition of being finite at all points. In other cases, the potential may be odd and the wave function must be odd in order to satisfy the boundary condition.

I hope this helps to clarify your questions. If you have further inquiries, feel free to ask. And yes, this discussion may be better suited for the homework section. Best of luck with your studies.
 

1. What is the Schrodinger Equation?

The Schrodinger Equation is a mathematical equation used to describe the quantum behavior of particles, such as electrons. It was developed by Austrian physicist Erwin Schrodinger in 1926.

2. What does the Schrodinger Equation predict?

The Schrodinger Equation predicts the probability of finding a particle in a certain location at a specific time. It also describes the wave-like nature of particles and how they can exist in multiple states at the same time.

3. How is the Schrodinger Equation used in quantum mechanics?

The Schrodinger Equation is one of the fundamental equations of quantum mechanics and is used to solve for the wave function of a particle. The wave function can then be used to calculate various properties of the particle, such as its energy and position.

4. Is the Schrodinger Equation limited to a certain type of particle?

No, the Schrodinger Equation can be applied to any type of particle, including atoms, molecules, and subatomic particles. It is a universal equation that describes the quantum behavior of all particles.

5. What are the implications of the Schrodinger Equation?

The Schrodinger Equation has revolutionized our understanding of the behavior of particles at the quantum level. It has led to the development of various technologies, such as transistors and lasers, and has allowed scientists to make accurate predictions about the behavior of matter on a subatomic level.

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