Particle in a box
The particle in a box, also called infinite potential well is a special case of the potential well, in which the potential in a certain area equal to zero and outside thereof is infinite. The particle in a box is a model system in quantum mechanics, which makes the quantization of the energy of course. As a one-dimensional model, it can be relatively easy to calculate. However, it is a simplifying assumption that does not completely conform to the rules of quantum mechanics, because the resulting wave function eg not continuously differentiable is ( at the junction of the potential barrier and box interior ).
- 5.1 Separation Approach
- 5.2 One-Dimensional Problems
- 5.3 overall solution
- 5.4 degeneration
- 6.1 Separation Approach
- 6.2 ball Symmetric Solutions
- 6.3 Non - spherically symmetric solution
Structure and Requirements
The one-dimensional model system consists of a free particles, for example, a gas molecule, which is in the floating space between two infinitely large potentials. Those referred to as " walls " limits ( at a and at a ) are perpendicular to the x -axis and thus parallel to each other. This simplistic model of a potential well is referred to as potential box.
Within the potential box of length no forces act on the particle in the model ( gravitational and electromagnetic fields are not taken into account). Since the potential outside the box is infinitely large, the particles can not leave the box. It follows that the particles in the interior of the box is moving with constant speed, and is reflected on the walls without loss of energy. Considering as a vector quantity, the latter shall indicate that the magnitude of the velocity remains constant.
State function and Antreffwahrscheinlichkeit
Describing the particle, as in quantum physics usual, with the help of a simple wave function, it follows that the interior of the potential box can exist only those particles for which is an integer multiple of half its wavelength, because only here also falls after reflection wave crest on wave crest and trough to trough ( standing wave ). Is not a multiple of half the wavelength, the shaft clears by superimposing a short time itself. This is the first feature of quantum mechanics, which can be illustrated with the particle in a box: particles within a potential well may exist that are described by the quantum number unique only in certain states.
An additional quantum mechanical feature in the model is the Antreffwahrscheinlichkeit, thus the probability of finding a particle at a specific location. The probability of finding the particle somewhere in the potential well is, because it can not leave the box. Everywhere outside the box is the Antreffwahrscheinlichkeit accordingly. For individual points within the box the Antreffwahrscheinlichkeit is different and depends on the state of the particle.
Another special feature of quantum mechanics, the tunneling effect does not occur in the described potential here, but only at a finite high potential well.
Energy
Because particles can exist only in certain individual states within a potential box, they can also only certain discrete energy values are dependent of. This also applies when at last high " walls " and has far-reaching implications as to the understanding of the structure of atoms. With the assumptions made above can be derived for the energy of a particle as a function of the following equation:
If a particle is excited, ie about one atom by irradiation energy is supplied, it switches without "floating " transition directly to a higher energy level ( "quantum leap" ). Moves a particle to a lower energy level, it emits the energy released, for example in the form of a photon.
From the above equation, there are three simple conclusions can be drawn that describe the particle in the potential box quality:
These statements also apply mutatis mutandis for other potential wells.
The solutions of the Schrödinger equation lead to the quantization of energy
The Hamiltonian of the one-dimensional problem is in position representation
The Schrödinger equation
Goes with the approach
In the time-independent (stationary ) Schrödinger equation over.
In the following, the time-independent Schrödinger equation is to solve his ( eigenvalue problem of the Hamiltonian )
Inside the box
The stationary Schrödinger equation corresponds inside the box of a free particle ( ordinary differential equation of 2nd order )
For the wave function inside the box to choose the following approach
Equivalent would be the approach with complex exponential functions.
This approach is set into the Schrödinger equation, the second derivative with respect to the place.
Thus is obtained the power as a function of the wave number:
Outside the box, continuity condition
Outside the box, the wave function must be identically zero due to the infinitely high potential.
However, since the wave function must be continuous throughout, thus conditions are imposed on the wave function in the box, namely, that the wave function of the walls is equal to 0:
Boundary condition 1
From the first boundary condition follows for the wave function inside the box
In order for this equation is satisfied, it must be. Thus, the wave function simplifies to
Boundary condition 2
With the aid of the second boundary condition is then followed for the wave function inside the box
In order for this equation is satisfied, it must be a whole multiple of ( the trivial solution A = 0 would mean that no wave exists), so
Thus, the wave number can assume only discrete values
Actually, it follows from the second boundary condition only that is an integer. For, however, the wave function would be zero everywhere and thus does not satisfy the normalization condition, that is not allowed. For negative, the wave function is up to sign the same as for the positive, namely. Since wave functions that differ by a factor describing the same state, bring the negative integers produced no new states. Therefore, it is limited to
As calculated above, the power depends on the wave number, supplies used:
Since only integer values can be assumed, the energy can also only assume certain values. The energy of the particle is thus quantized, the energy levels are "discrete".
Standardization
The amplitude can still be determined via the normalization condition:
Since a complex number, only their sum is fixed, the phase is arbitrary:
Wave functions, which differ only by a constant phase factor, describe the same state. Therefore, you can set and thus choose real.
Summary
The eigenvalues ( = potential energy values) and eigenfunctions ( = wave functions ) of the Hamiltonian for a particle in a box with infinitely high potential walls are thus:
Ground state
The ground state energy (lowest possible energy ) is not zero ( because of the Heisenberg uncertainty principle not allowed), but
This can also be obtained from consideration of the Heisenberg uncertainty principle: the particle is restricted to the region of space. Then the minimum pulse yields over. Within the box, the potential is zero, so the total energy is equal to the kinetic energy.
Three-dimensional case ( square )
In the three-dimensional box ( rectangular ) of the Hamiltonian looks as follows:
In this case, the potential
The full Hamiltonian can be by means of
As a sum of three one-dimensional Hamiltonians write:
Separation of variables
The stationary Schrödinger equation ( three-dimensional)
Can be combined with the following product approach
Separate into three one-dimensional problems.
Set to the product approach in the stationary Schrödinger equation, and use of, that only acts, that is, the other one can pass by the Hamiltonian.
Share by delivering:
Thereby separating the three constants, , defining the sum of which the total energy:
One-Dimensional Problems
Now you have for each spatial direction separately the one-dimensional problem, as has been done above, dissolved:
Their solution is:
Overall solution
The solution of the three-dimensional box is for the total wave function of the product of the one-dimensional wave functions
And the total energy of the sum of the one-dimensional energy eigenvalues :
Degeneration
The energy eigenvalues may be degenerate, that is, different wave functions have the same energy. This means that the three-dimensional case that different quantum numbers lead to the same sum.
For example, the case for a special case of the cube, ie, degeneracies on. The energy is given by:
For degeneracy different quantum numbers must lead to the same sum.
The lowest energy level is not degenerate ( = single degenerate ) and thus.
The next higher energy value is already degenerate threefold: thus and.
It can also occur higher degeneracies than three times, for example Thus 4-fold and.
Three-dimensional case (ball )
For the three-dimensional spherical box with radius, it is useful to represent the Hamiltonian in spherical coordinates:
In this case, the potential
Separation of variables
Just as in the hydrogen atom can separate the Schrödinger equation into two independent equations, where the wave function is derived from the product of a radius-dependent function and the spherical harmonics:
It is here the main or energy quantum number, the angular momentum quantum number and the magnetic quantum number.
For the radius-dependent function following radial Schrödinger equation remains ( where V = 0 has been taken into account within the box ):
A solution is found to be due to the angle-dependent Schrödinger equation:
Ball Symmetric Solutions
Initially, only the simple case will be considered ( s-like wave functions ). Thus, the term vanishes from the radial Schrödinger equation.
Additionally, it should be set. It follows:
Thus, the radial Schrödinger equation simplifies to:
As is immediately apparent, is the solution for the same as for particles in the linear case: or
R (0) must have a finite value, so that the term is eliminated. In addition, the boundary condition R ( L ) = 0 because of the continuity of the wave function. It follows for k:
Use of U (r) in the radial Schrödinger equation yields:
From which can be determined with the energy eigenvalues .
In summary: For ( spherically symmetric solutions ), the wave functions arise with the normalization constant and the energy eigenvalues to:
Non - spherically symmetric solution
For l > 0, the solution of the Schrödinger equation is much more complicated. On their spherical Bessel functions jl, associated with the normal Bessel functions Jl following result:
Hangs due to the boundary condition on the respective square -driven nth zero of these features:
Which are not analytically determined.
Model for conjugated systems
The particle in a box can be a simple model for a conjugated molecule, eg Hexatriene, can be used to estimate the energy. It is assumed that the electrons can move freely in a conjugated molecule in this, but it can not leave. Adding formally half an atom at each end of the molecule. The length of this particle then corresponds to the case where the electron is.
Examples
An example from the crystallography is the color center in which an electron is trapped in an anion vacancy and can be described to a good approximation as a particle in a box. Also the color of dyes with linear conjugated pi - systems can capture up by considering the Pi - system as a one-dimensional particle in a box problem.