What is the momentum of an ultraviolet photon with a frequency of 1.50 petahertz? 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.
To answer this question, we will need to know how to relate the momentum of a photon to its frequency. This relationship can be expressed in terms of the Planck constant and the speed of light in vacuum as the momentum of the photon is equal to the Planck constant times the photon’s frequency divided by the speed of light in vacuum. Since we only need two decimal places of accuracy in our final answer, we can report the speed of light as 3.00 times 10 to the eighth meters per second.
Also, since we are looking for a final answer in scientific notation, let’s rewrite the frequency of 1.50 petahertz in terms of just hertz and scientific notation. To do this, we recall that one petahertz is 10 to the 15th hertz. One helpful way to remember this number is to remember the hierarchy of SI prefixes, which go giga-, tera-, and then peta-. So since giga- is 10 to the ninth, tera- would be 10 to the 12th, and peta- would be 10 to the 15th. So the frequency of our photon is 1.50 times 10 to the 15th hertz.
We now have a value for the speed of light, the frequency, and also the Planck constant. So all that’s left is for us to substitute these values into the formula. So we have 6.63 times 10 to the negative 34 joule-seconds times 1.50 times 10 to the 15th hertz divided by 3.00 times 10 to the eighth meters per second. The units of this quantity may look a little bit daunting. We have joule-seconds times hertz divided by meters per second. But remember, the overall units must agree with momentum.
Hertz are inverse seconds. In other words, they can be expressed directly as SI base units. Similarly, joules are kilograms meters squared per second squared. So joules can also be expressed directly as SI base units. Seconds and meters are themselves SI base units. Now, remember, SI base units are constructed so that any combination of them gives another combination of SI base units without introducing any additional numeric factors. The SI base units for momentum are kilogram-meters per second. So based on what we just talked about, joule-seconds times hertz divided by meters per second must be kilogram-meters per second. And we know this even without going through any mathematics.
Putting the rest of the numbers into a calculator, we find that the momentum of this ultraviolet photon is 3.315 times 10 to the negative 27 kilogram-meters per second. Rounding to two decimal places gives us our final answer, 3.32 times 10 to the negative 27 kilogram-meters per second.