Particle in a box The particle in a box is one of the simplest but also the most important models in quantum mechanics.

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Particle In A Box The Particle In A Box Is One Of The Simplest But Also The Most Important Models In Quantum Mechanics 1 (409.08 KiB) Viewed 51 times

Particle In A Box The Particle In A Box Is One Of The Simplest But Also The Most Important Models In Quantum Mechanics 2 (320.03 KiB) Viewed 51 times

Particle in a box The particle in a box is one of the simplest but also the most important models in quantum mechanics. It consists of an m- mass particle stuck in a potential well. The value of the potential energy inside the box is zero but outside it is infinite. Such a potential ensures that the particle remains inside the box and that the wave function of the particle is thus exactly zero at all points outside the box, otherwise the energy of the particle would be infinite. In addition, the potential is assumed to be constant inside the box. The simplicity of the model allows for an accurate solution. Although a particle in an infinite potential pit may sound like a very unrealistic situation, it can be used to explain a little bit how, for example, a person's sense of sight, and especially color, works. There is an important light-absorbing molecule in the human eye called the retinal. The retinal molecule consists of a chain of carbon atoms and is approximately 1,5.10^-9 m long. The electron in this long molecular chain behaves much like a particle in a box. This task explores what we can learn about the function of the retinal molecule with a simple ‘particle in a box' model. The box itself is only a concrete analogy to boundary conditions that constrain a particle to some area of space, for example, for x € [0, L] in one dimension x E [0, L). In principle, we can define the potential energy function U (x) elsewhere but in practice this potential energy function is irrelevant if we just force the right boundary conditions on the wave function of the particle. For a particle in a box, these boundary conditions are that the wavefunction y(x) is zero for all x # [0, L]. In addition, since the wavefunction must be continuous, its values must be at the edges of the box y(O) = y(L) = 0. The time-dependent Schrödinger equation for the wave function y(x) inside the box (i.e. when x E [O, L]) is au (x, t) ħ az ih Î14 (x, t) , where în However, it is always useful to first solve the eigenstates of at 2т дх2 — — the operator H These are the states On (2) with well-defined energies Em that implement the eigenvalue equation Hon (3) ñ a?on (x) = En On () 2m дх2 a.) Solve the solutions of this eigenvalue equation On (2) (eigenstates) and the corresponding energies En (specific energy). b.) Consider a free electron inside the retinal molecule and how the electron excites as it absorbs the photon. It is imagined that the electron is initially in the eigenstance on (x) for some arbitrary integer n. Determine the wavelength in that a photon must have so that the electron that absorbs it can excite from state n to the next state n+1. Assume that the length of the box is L and the electron has mass m.
c.) Check and verify that as the quantum number n increases, the required photon wavelength a decreases. This means that the energy of the photon has to increase, which makes sense because the energy differences between the energy levels of the box increase as the quantum number n increases d.) Make a short magnitude calculation to show that the wavelength of the photon required for an electron in its ground state (n= 1) in a box of length 10^-9 m is close to the wavelengths of visible light (it is slightly on the infrared side). 4mcL2 1 Using the results you get for resonant wavelengths In = , a slightly correct retinal molecule is πή 2n +1 analyzed. The molecule has twelve electrons that can move freely along the carbon atom chain. Due to Pauli exclusion principle, these twelve electrons fill the lowest six spaces in a box, two electrons in each state. Thus, the smallest possible photon energy that can be absorbed would be one that excites an electron from the sixth state to the seventh state. e.) Use the result above to solve the numerical value for the wavelength of the smallest excitation photon. The length of the retinal molecule is 1.5 . 10^-9 m. f.) What color (visible light) does this wavelength of photon correspond to? g.) Determine the optimal potential well length at which the retinal molecule would be as sensitive as possible to blue, green, and red light as shown in the figure: 420 nm 498 nm 534 nm Green cones 564 nm Red cones Blue cones Rods 100 Normalized absorbance 50 - Short Medium Long 400 Violet TUTT TTTTTTTTTT 500 600 700 Cyan Green Yellow Red Blue Wavelength (nm)