What is the energy of a photon that has a frequency of 5.50 times 10 to the 14th hertz? Use a value of 6.63 times 10 to the negative 34 joule-seconds for the Planck constant. Give your answer in scientific notation to two decimal places.
This question asks us to relate the energy of a photon to its frequency, and we suspect that that relationship will involve the explicit value that we are given for the Planck constant. Indeed, the energy of a photon is given by the formula 𝐸 is equal to ℎ𝑓, where 𝐸 is the energy of the photon, ℎ is the Planck constant, and 𝑓 is the photon’s frequency. With this formula, we see that 5.50 times 10 to the 14th hertz, the frequency of our photon, times 6.63 times 10 to the negative 34 joule-seconds, the value we’re given for the Planck constant, will give us the energy that we are looking for. 5.50 times 6.63 is 36.465. So in scientific notation, we have 3.6465 times 10 to the negative 19 hertz joule seconds.
Now recall that hertz are defined as inverse seconds, and the relationship is one to one. One hertz is one inverse seconds. This means that hertz combines with seconds to give a dimensionless quantity because hertz-seconds is just seconds per second, which is one. We therefore find that our overall units are joules, which is good because we are looking for an energy and joules are a unit of energy. The last thing we need to do is to round our answer to two decimal places, and we find that the energy of our photon is 3.65 times 10 to the negative 19 joules.
When we work with photons, it is often convenient to express their energies in terms of electron volts. This energy is equivalent to about 2.3 electron volts. 2.3 electron volts is a little bit more energetic than the energy difference between the second and third excited states of the hydrogen atom. Although to answer this question we don’t need to know anything about the hydrogen atom or the energy of the photon in electron volts, thinking about these relationships helps us build up an intuition for the relevant energy scales in atomic and nuclear physics.