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physics 2

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- Matter is made of charged particles

- The motion of charged particles produces electromagnetic waves

- Since light is an electromagnetic wave, the color of a substance is ultimately related to the way

that charged particles in that substance can move around. Quantum mechanics tells us that:

An electron (or any sub-atomic particle) that moves with momentum (mass times

velocity) acts like a wave with wavelength , where is Planck%u2019s constant.

These %u2018matter waves%u2019 can exist in certain %u2018states%u2019, analogous to standing waves, in which

they do not radiate.

The %u2018motion%u2019 that produces- or absorbs- radiation occurs when an electron jumps between

different states.

The difference in energy of these states, , is equal to the energy of the radiation.

The light comprising this radiation obeys the Planck/Einstein formula: , where is

the frequency.

Shown below is a molecule known as beta-carotene:

This molecule is found in carrots and is responsible for their orange color. We will now see why.

This molecule consists of a long straight chain of carbon atoms (black spheres), bonded to hydrogen atoms (white spheres). This molecule can be thought of as a long string, and the electron states correspond to standing waves on that string.

The goal here is to find the different energies corresponding to these standing waves. The energy differences between standing wave states will then determine the possible energies of absorbed photons.

First, we start with some background information. The length of this carbon chain is about L=18.5 x 10^-10 m
m, and there are 22 electrons. Due to the exclusion principle, only one electron

can be in each state. However, because of electron spin, which has two states, each standing wave can accommodate 2 electrons. Therefore, we need 22/2 = 11 standing wave states to accommodate all these electrons. Energy can be absorbed when an electron jumps into the next highest (i.e. the 12th) state.

Thus, we want to figure out the energies corresponding to the 11th and 12th standing wave states, and set that equal to the absorbed photon energy.

The procedure is:

1. Determine the wavelengths of standing waves on the molecule or 'string'. The only rule is that the wave amplitude must be zero at either end. This is exactly the same as for an actual string, so the formula for lambda is the same as given in Chapter 12 of the textbook.

2. Once you know , lambda use p=mv=h/(lambda) to determine the momentum of the electron.

3. The total energy is the sum of kinetic and potential energies. In this model we assume that,
since the molecule has no net charge, the forces due to all other protons and electrons cancel
out. Therefore, the electron has kinetic energy only, equal to E=(1/2)mv^2 = (p^2)/2m

4. Find the difference in energy between the 12th and 11th states, ∆E.

5. Set ∆E=hf to determine the frequency of a photon that can cause an electron to change
states. Also calculate the wavelength of the photon.

6. Determine what color of light this wavelength corresponds to. This is the color that the
molecule absorbs. The color that we see corresponds to what is not absorbed.

please show all calculations and explain fully thanks.

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