What is Schrodinger's equation? How to solve it? What is the meaning of symbol si in it?
Answers
The Schrödinger equation is a linear partial differential equation that describes the wave function or state function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject
It is generally solved using the Numerov method, which is a powerful approach to integrate second-order differential equations of the general form
Schrodinger’s equations tells how particles move through space and time. In particular how small particles move. However, rather than being an equation that allows you to compute the trajectory of the particle, it computes a probability distribution of finding the particle at a particular location at a particular time. Since very small particles typically do not have an exact location at any given time, Schrodinger’s equation just computes a probability function (or more accurately the wave function) from which you can compute the probability of finding the particle. As an aside, one can derive the uncertainty principle from the Schrodinger’s equation (which says says that the product of the uncertainty of the the position and momentum of a particle or of the energy and time associated with a particle is always greater than plank’s constant). For larger particles or groups of particle (e.g., like a grain of sand), in principle the equation still applies, but the uncertainty is so small that it can be ignored for all practical purposes.
Regarding a little more detail on the how, Schrodinger’s equation only compute the probability distribution, one needs to average the quantity that one is interested in weighted by the probability distribution.
The quantity that is computed by Schrodinger’s equation is called a wave function, because the probability function travels like a wave. All particles have wave like characteristics. For example, the momentum of a particle is equal to a constant divided by the wavelength of the particle.
Now, lets say you want to compute the position of a particle based on the wave function psi(x,t) generated from Schrodinger’s equation, then one take the complex conjugate of the wave function psi*(x,t) multiplies it by the position x, and by psi(x,t), and the integrates ((psi*(x,t)(x)psi(x,t)) over the position to find the most likely position of the particle at any given time (or the expectation value for the position). As an overly simplified example, of how the wave function gives an expectation value for a quantity, the wave function, computed by Schrodinger’s equation may be
psi*(x=1)-psi(x=1)=.5,
psi*(x=2)=psi(x=2)=sqrt(.5),
psi*(x=3)=psi(x=3)=.5
(not a very realistic wave function, but it make the math easier)
Then the wave function tells you that a particular particle has a
psi*(x=1)psi(x=1)=.25
probability of being at position x=1, a
psi*(x=2)psi(x=2)=.5
probability of of being at position x=2 and a
psi*(x=3)psi(x=3)=.25
probability of being a position x=3 and cannot be found anywhere else. Then, the integral of (psi*(x,t)(x)psi(x,t), would be
.25(1)+.5(2)+.25(3)=2, meaning that position 2 is the most likely position where the particle is found.
Note that psi(x,t) is chosen (e.g., by dividing the function associated with psi(x,t)) so that (in addition to satisfying Schrodinger’s equation) the integral over all positions of (psi*(x,t)psi(x,t))=1, indicating that there is a 100% probability of finding the particle somewhere - at at-least one of the possible positions.
Answer:
- Schrodinger’s equations tells how particles move through space and time.
- In particular how small particles move.
- However, rather than being an equation that allows you to compute the trajectory of the particle, it computes a probability distribution of finding the particle at a particular location at a particular time. Since very small particles typically do not have an exact location at any given time, Schrodinger’s equation just computes a probability function (or more accurately the wave function) from which you can compute the probability of finding the particle. As an aside, one can derive the uncertainty principle from the Schrodinger’s equation (which says says that the product of the uncertainty of the the position and momentum of a particle or of the energy and time associated with a particle is always greater than plank’s constant). For larger particles or groups of particle (e.g., like a grain of sand), in principle the equation still applies, but the uncertainty is so small that it can be ignored for all practical purposes.