Schrodinger's Equation

Contents Home Statistical Interpretation Schrödinger's Equation Heisenberg's Uncertainty Principle Pauli Exclusion Principle Credits Sources	Schrödinger's Equations Recall from the section on Heisenberg's Uncertainty Principle that particles at the microscopic scale can be described by a quantum wave function . This isn't necessarily a function of space. It can be a function of momentum instead, and either case will provide an adequate description of the function. The state vector is a concept used in Dirac notation that represents the "constant" behind the wave. The advantage of using vectors is that one can apply operators to them through matrix algebra. In other words, the state vector completely describes the state of a physical system. The Schrodinger Equations are partial differential equations that can be solved in order to find the form of a particle's wave function under certain conditions. So, if the particle experiences a known potential, and is limited to a certain region of space, then we can determine the nature of it's wave function. The Time Dependent Schrodinger Equation governs the form of the state vector as a particle's physical system changes over time. It is as follows: This is obviously just for the one-dimensional case were the particle may only move along the one axis, but can be adapted to higher dimensions by substituting the laplacian operator for (d^2/dx^2) - this represents the second-order derivatives with respect to y and z. Here "V" represents the potential experienced by the particle as a function of position. The function of on the left hand side is also known as the Hamiltonian operator. The total energy of any system involving quantum particles can be determined by applying this operator to the wavefunction of that particle. The time-dependant function on the right hand side tells us that the energy of the system will vary with time. Various other operators maybe applied to find the momentum of the particle - given by i(d/dx) and it's kinetic energy - given by - - among other quantities. The Time Independent Schrodinger Equation governs the form of the state vector when it doesn't change with time. It is as follows: One consequence of using the TISE is that all the systems it describes have constant energy. We can see this because the Hamiltonian operation on the state vector results in an eigenvalue E appearing on the left hand side. One other condition on wavefunctions is that they must be normalised over the space in which the particle may appear, in order that the probability of the particle appearing somewhere within the required space is always 1. Usually this is over all space, but sometimes (as in the case of an infinite square potential well) it may be less. A wavefunction must be normalised over the required space in order for the results of operators to make sense. The Schrodinger Equation isn't the only one that makes use of wavefunctions. The Klein-Gordon Equation (in which space and time are both second-order derivatives) was actually formulated before hand. Schrodinger didn't like it because it only worked for spinless (spin = 0), charge-neutral particles. However in changing it Schrodinger lost it's relativistic correctness - this is because it treats time and space differently (by giving them different derivatives). The next "evolution" in quantum equations was the Dirac equation, which aside from being linear (containing only first order derivatives), did account for spin-1/2 particles (such as electrons). It was almost delightfully simple (asked "How did you find it?" Dirac responded, "Beautiful"). It is as follows: Another advantage of the Dirac equation is that it accounts for atomic light-emission spectra in ways that non-relativistic quantum mechanics (such as the Schrodinger Equation) was unable to. A third consequence of the Dirac equation (one that we won't derive here) is particularly compelling. It was known by this time that electrons possessed "spin", or angular momentum. In fact, this had to be assumed for most other quantum mechanical theories to work. This was not required of the Dirac equation - in fact spin (and as an added bonus, the magnetism) of electrons became a natural consequence of the combination of quantum and relativistic properties. Thus Dirac explained the reason why such properties existed. Spin and Magnetism Unlike the wavefunctions input into the Schrodinger equations, the Dirac equation only accepts input from state vectors with four components, one of which was the "spin" mentioned earlier. But the intriguing thing was this: for every solution of the Dirac equation for a particle with positive energy there was a counterpart solution requiring the existence of negative energy. In the case of the electron (+ve energy) Dirac called it's -ve energy equivalent the "anti-electron", or as it is more widely known, the positron. Thus Dirac predicted the existence of anti-matter, and the positron itself was discovered experimentally in 1932. Antimatter But what of ground-states (minimum energy)? These are known to exist and be positive non-zero. Dirac, after much puzzling, put forward the theory that there existed a "sea" of particles with negative energy that "filled up" all the negative energy states. Electrons then "floated" on this sea, moving towards the lowest energy states when possible. This is what happens when a positron and electron collide: the electron loses energy as it moves to a negative-energy state, and this is emitted in the form of photons. So when a positron and electron meet, they "annihilate" each other as the electron disappears into the Dirac sea. top

Schrödinger's Equations

Recall from the section on Heisenberg's Uncertainty Principle that particles at the microscopic scale can be described by a quantum wave function . This isn't necessarily a function of space. It can be a function of momentum instead, and either case will provide an adequate description of the function. The state vector is a concept used in Dirac notation that represents the "constant" behind the wave. The advantage of using vectors is that one can apply operators to them through matrix algebra. In other words, the state vector completely describes the state of a physical system.

The Schrodinger Equations are partial differential equations that can be solved in order to find the form of a particle's wave function under certain conditions. So, if the particle experiences a known potential, and is limited to a certain region of space, then we can determine the nature of it's wave function.

The Time Dependent Schrodinger Equation governs the form of the state vector as a particle's physical system changes over time. It is as follows:

This is obviously just for the one-dimensional case were the particle may only move along the one axis, but can be adapted to higher dimensions by substituting the laplacian operator for (d^2/dx^2) - this represents the second-order derivatives with respect to y and z. Here "V" represents the potential experienced by the particle as a function of position. The function of on the left hand side is also known as the Hamiltonian operator. The total energy of any system involving quantum particles can be determined by applying this operator to the wavefunction of that particle. The time-dependant function on the right hand side tells us that the energy of the system will vary with time. Various other operators maybe applied to find the momentum of the particle - given by i(d/dx) and it's kinetic energy - given by - - among other quantities.

The Time Independent Schrodinger Equation governs the form of the state vector when it doesn't change with time. It is as follows:

One consequence of using the TISE is that all the systems it describes have constant energy. We can see this because the Hamiltonian operation on the state vector results in an eigenvalue E appearing on the left hand side.

One other condition on wavefunctions is that they must be normalised over the space in which the particle may appear, in order that the probability of the particle appearing somewhere within the required space is always 1. Usually this is over all space, but sometimes (as in the case of an infinite square potential well) it may be less. A wavefunction must be normalised over the required space in order for the results of operators to make sense.

The Schrodinger Equation isn't the only one that makes use of wavefunctions. The Klein-Gordon Equation (in which space and time are both second-order derivatives) was actually formulated before hand. Schrodinger didn't like it because it only worked for spinless (spin = 0), charge-neutral particles. However in changing it Schrodinger lost it's relativistic correctness - this is because it treats time and space differently (by giving them different derivatives).

The next "evolution" in quantum equations was the Dirac equation, which aside from being linear (containing only first order derivatives), did account for spin-1/2 particles (such as electrons). It was almost delightfully simple (asked "How did you find it?" Dirac responded, "Beautiful"). It is as follows:

Another advantage of the Dirac equation is that it accounts for atomic light-emission spectra in ways that non-relativistic quantum mechanics (such as the Schrodinger Equation) was unable to. A third consequence of the Dirac equation (one that we won't derive here) is particularly compelling. It was known by this time that electrons possessed "spin", or angular momentum. In fact, this had to be assumed for most other quantum mechanical theories to work. This was not required of the Dirac equation - in fact spin (and as an added bonus, the magnetism) of electrons became a natural consequence of the combination of quantum and relativistic properties. Thus Dirac explained the reason why such properties existed.

Spin and Magnetism

Unlike the wavefunctions input into the Schrodinger equations, the Dirac equation only accepts input from state vectors with four components, one of which was the "spin" mentioned earlier. But the intriguing thing was this: for every solution of the Dirac equation for a particle with positive energy there was a counterpart solution requiring the existence of negative energy. In the case of the electron (+ve energy) Dirac called it's -ve energy equivalent the "anti-electron", or as it is more widely known, the positron. Thus Dirac predicted the existence of anti-matter, and the positron itself was discovered experimentally in 1932.

Antimatter

But what of ground-states (minimum energy)? These are known to exist and be positive non-zero. Dirac, after much puzzling, put forward the theory that there existed a "sea" of particles with negative energy that "filled up" all the negative energy states. Electrons then "floated" on this sea, moving towards the lowest energy states when possible. This is what happens when a positron and electron collide: the electron loses energy as it moves to a negative-energy state, and this is emitted in the form of photons. So when a positron and electron meet, they "annihilate" each other as the electron disappears into the Dirac sea.

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Schrödinger's Equations

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