what what do you understand by the term sine function and sine value as used while solving Schrodinger Wave Equation
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There are whole books written on this, so I can just give an overview, and I’ll do it without writing any equations.
When the Schrödinger equation was written, it was already known that matter is made of waves. De Broglie had predicted it and the Davisson-Germer experiment had confirmed that, showing that electrons could be diffracted. It was therefore urgent to describe these waves.
So the first requirement of Schrödinger equation was that its solutions should be waves. But these solutions had to be different from those of Maxwell’s equations or the telegrapher’s equations. The wave solutions of those equations have a fixed propagation speed, but the speed of the waves for the new equation had to depend on the velocity of the electron being described. Therefore Schrödinger arranged that his equation has dispersion, the property that the velocity of waves is frequency dependent.
In order to describe an electron that is confined to a small region, it is logical that its wave function be nonzero only in that region. (The idea that the wave function represents a probability was developed later by Born). But if we represent a localized wave function as a sum of single-frequency solutions, we can see that it contains a wide variety of frequencies. The equation is linear, which allows the different frequency components to travel at different velocities without affecting each other. The initially localized wave function will disperse over time. This is related to the uncertainty principle.
In addition to the wavelike solutions, there are also bound states. The nucleus of the atom has a large positive charge, which traps negatively charged electrons in “orbits”. The Schrödinger equation does a fairly good job of predicting the orbits.
In this case, we represent the force of the charged nucleus as a potential energy well. The potential energy at any point is added onto the kinetic energy in the Schrödinger equation, and this causes the electron to respond to a force equal to the gradient of the potential.
The “orbits” of electrons correspond to time-independent solutions of the equation, each with a given energy. This is commonly interpreted to mean that the electron can only be in an orbit corresponding to one of these solutions. But more precisely, the general solution of the wave function is the sum of these particular solutions. A general solution corresponds to an electron with uncertain energy, and will not necessarily be time-independent. This is all related to the atom exchanging energy with its surroundings via the electromagnetic field.
The Schrödinger equation is a first attempt at describing electrons, and has various difficulties. For example, even a slowly moving observer should logically see a completely different wave function from an observer at rest, because the solutions are velocity dependent. Moving the observer around simply breaks it. More importantly, the results of actual experiments differ from the equation’s predictions.
The first improvement was by Pauli, in response to the Stern-Gerlach experiment. He added an extra component to the wave function to properly describe electron spin. Next Dirac added two more components to make the equation compatible with relativity, and in doing so got near-perfect agreement with experiment.
Read Schiff’s Quantum Mechanics for the details. Physics Stack Exchange is the better forum if you have questions after that.
it's not a perfect answer but I hope you like it ,and it will help you
When the Schrödinger equation was written, it was already known that matter is made of waves. De Broglie had predicted it and the Davisson-Germer experiment had confirmed that, showing that electrons could be diffracted. It was therefore urgent to describe these waves.
So the first requirement of Schrödinger equation was that its solutions should be waves. But these solutions had to be different from those of Maxwell’s equations or the telegrapher’s equations. The wave solutions of those equations have a fixed propagation speed, but the speed of the waves for the new equation had to depend on the velocity of the electron being described. Therefore Schrödinger arranged that his equation has dispersion, the property that the velocity of waves is frequency dependent.
In order to describe an electron that is confined to a small region, it is logical that its wave function be nonzero only in that region. (The idea that the wave function represents a probability was developed later by Born). But if we represent a localized wave function as a sum of single-frequency solutions, we can see that it contains a wide variety of frequencies. The equation is linear, which allows the different frequency components to travel at different velocities without affecting each other. The initially localized wave function will disperse over time. This is related to the uncertainty principle.
In addition to the wavelike solutions, there are also bound states. The nucleus of the atom has a large positive charge, which traps negatively charged electrons in “orbits”. The Schrödinger equation does a fairly good job of predicting the orbits.
In this case, we represent the force of the charged nucleus as a potential energy well. The potential energy at any point is added onto the kinetic energy in the Schrödinger equation, and this causes the electron to respond to a force equal to the gradient of the potential.
The “orbits” of electrons correspond to time-independent solutions of the equation, each with a given energy. This is commonly interpreted to mean that the electron can only be in an orbit corresponding to one of these solutions. But more precisely, the general solution of the wave function is the sum of these particular solutions. A general solution corresponds to an electron with uncertain energy, and will not necessarily be time-independent. This is all related to the atom exchanging energy with its surroundings via the electromagnetic field.
The Schrödinger equation is a first attempt at describing electrons, and has various difficulties. For example, even a slowly moving observer should logically see a completely different wave function from an observer at rest, because the solutions are velocity dependent. Moving the observer around simply breaks it. More importantly, the results of actual experiments differ from the equation’s predictions.
The first improvement was by Pauli, in response to the Stern-Gerlach experiment. He added an extra component to the wave function to properly describe electron spin. Next Dirac added two more components to make the equation compatible with relativity, and in doing so got near-perfect agreement with experiment.
Read Schiff’s Quantum Mechanics for the details. Physics Stack Exchange is the better forum if you have questions after that.
it's not a perfect answer but I hope you like it ,and it will help you
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