the “length of the box,” i.e. the radius of the quantum dot. Manipulation of the band gap allows for the absorption and emission of specific wavelengths of light, as energy is inversely proportional to wavelength. The smaller the quantum dot, the larger the band gap and thus the shorter the wavelength absorbed. Different semiconducting materials are used to synthesize quantum dots of different sizes and therefore emit different wavelengths of light. Materials that normally emit light in the visible region are often used and their sizes are fine-tuned so that certain colors are emitted. Typical substances used to synthesize quantum dots are cadmium (Cd) and selenium (Se). For example, when the electrons of two nanometer CdSe quantum dots relax after excitation, blue light is emitted. Similarly, red light is emitted in four nanometer CdSe quantum dots. Quantum dots have a variety of functions including but not limited to fluorescent dyes, transistors, LEDs, solar cells, and medical imaging via optical probes. 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. Quantum dots are useful for these functions due to their emission of brighter light, excitation by a wide variety of wavelengths, and higher resistance to light than other substances. A more general model: particle in a box with a period potential A more general model is the particle in a box with a period potential model: The box's interior has a periodic potential, and the box contains periods of a positive integer number in each dimension. A periodic potential becomes a constant potential, and a Bloch wave becomes a plane wave when the period(s) involved goes zero. The periodic potential is more general than the constant potential; the Bloch wave is more general than the plane wave. A recent new theory investigated this model based on the mathematical theory of periodic differential equations. The problem can be considered quantum confinement of Bloch waves, more general than the quantum confinement of plane waves discussed in previous sections. The new theory found that, in one-dimensional cases, the problem can be analytically solved. Moreover, in many essential and simple multi-dimensional cases, the problem can be analytically solved based on relevant mathematical theorems with the help of reasonings of physics intuitions. In