Particle in a box
The particle in a box model is one of the very few problems in quantum mechanics that can be solved analytically, without approximations.
Due to its simplicity, the model allows insight into quantum effects without the need for complicated mathematics.
Here, the particle may only move backwards and forwards along a straight line with impenetrable barriers at either end.
[1] The walls of a one-dimensional box may be seen as regions of space with an infinitely large potential energy.
[2] This means that no forces act upon the particle inside the box and it can move freely in that region.
However, infinitely large forces repel the particle if it touches the walls of the box, preventing it from escaping.
[6] Here one sees that only a discrete set of energy values and wave numbers k are allowed for the particle.
If we set the origin of coordinates to the center of the box, we can rewrite the spatial part of the wave function succinctly as:
For the centered box (xc = 0), the solution is real and particularly simple, since the phase factor on the right reduces to unity.
In quantum mechanics, however, the probability density for finding a particle at a given position is derived from the wave function as
If relativistic wave equations are considered, however, the probability density not go to zero at the nodes (apart from the trivial case
For a superposition of states, the expectation value of the position will change based on the cross term, which is proportional to
, and hence the minimum kinetic energy of the particle in a box is inversely proportional to the mass and the square of the well width, in qualitative agreement with the calculation above.
These systems are studied in the field of quantum chaos for wall shapes whose corresponding dynamical billiard tables are non-integrable.
[12] The conjugated system of electrons can be modeled as a one dimensional box with length equal to the total bond distance from one terminus of the polyene to the other.
[14] Due to β-carotene's high level of conjugation, electrons are dispersed throughout the length of the molecule, allowing one to model it as a one-dimensional particle in a box.
Using the previous relation of wavelength to energy, recalling both the Planck constant h and the speed of light c:
This indicates that β-carotene primarily absorbs light in the infrared spectrum, therefore it would appear white to a human eye.
However the observed wavelength is 450 nm,[16] indicating that the particle in a box is not a perfect model for this system.
Due to their small size, quantum dots do not showcase the bulk properties of the specified semi-conductor but rather show quantised energy states.
[20] Researchers at Princeton University have recently built a quantum well laser that is no bigger than a grain of rice.
This relationship is a key component in quantum mechanical theories that include the De Broglie Wavelength and Particle in a box.
The double quantum dot allows scientists to gain full control over the movement of an electron, which consequently results in the production of a laser beam.
[22] They display quantum confinement in that the electrons cannot escape the “dot”, thus allowing particle-in-a-box approximations to be used.
[22] The smaller the quantum dot, the larger the band gap and thus the shorter the wavelength absorbed.
[22] Typical substances used to synthesize quantum dots are cadmium (Cd) and selenium (Se).
[22][24] For example, when the electrons of two nanometer CdSe quantum dots relax after excitation, blue light is emitted.
[25][22] Quantum dots have a variety of functions including but not limited to fluorescent dyes, transistors, LEDs, solar cells, and medical imaging via optical probes.
[22][23] One function of quantum dots is their use in lymph node mapping, which is feasible due to their unique ability to emit light in the near infrared (NIR) region.
Lymph node mapping allows surgeons to track if and where cancerous cells exist.
