Video Transcript
Which of the following formulas
shows the relation between the de Broglie wavelength of a particle 𝜆, its momentum
𝑝, and the Planck constant ℎ? (A) 𝜆 equals 𝑝 divided by ℎ. (B) 𝜆 equals ℎ𝑝 squared. (C) 𝜆 equals ℎ divided by 𝑝
squared. (D) 𝜆 equals ℎ divided by 𝑝. Or (E) 𝜆 equals ℎ squared 𝑝
squared.
The question is asking us to
identify the formula for the de Broglie wavelength of a particle. And we are told what our variables
mean. The wavelength of the particle is
𝜆, the momentum is 𝑝, and the Planck constant is ℎ. The easiest way to answer this
question correctly is to know the de Broglie formula by heart. And if we know this, we can
immediately identify (D) as the correct answer.
But if we don’t know this formula
by heart, we can still figure out that (D) is the correct answer by paying attention
to the dimensions of our quantities. 𝜆 is a wavelength, which
unsurprisingly has dimensions of length. We’ll use the capital letter 𝐿 to
represent length. And what it means for the
dimensions of 𝜆 to be length is that if we reported a value for 𝜆, we would use
units like meters or centimeters or kilometers. But we can never use units like
seconds or coulombs or amperes.
Similarly, momentum is mass times
velocity, and velocity is distance per time. So the dimensions of momentum are
mass times length divided by time. The capital letter 𝑀 means mass,
and the capital letter 𝑇 means time. And just as a matter of convention,
we write 𝑇 to the negative one power rather than one divided by 𝑇.
Finally, the Planck constant. We may recall that one common value
for the Planck constant is 6.626 times 10 to the negative 34 joules seconds. Joules are a unit of energy, and
seconds are a unit of time. So the dimensions of the Planck
constant are energy times time. To express energy in terms of the
dimensions that we’ve already used, length, mass, and time, we can recall the
formula for kinetic energy, one-half mass times velocity squared.
As we can see from this formula,
the dimensions of energy are the same as mass times the dimensions of velocity
squared. And we know the dimensions of
velocity are length per time. So the dimensions of energy are
mass times length squared per time squared. These dimensions are not just for
kinetic energy, but for any energy. So the Planck constant with units
of energy times time has dimensions of mass times length squared times time to the
negative one power.
Now that we know the dimensions of
each of the three quantities that we’re working with, we’re ready to work out the
answer to the question. First, we observe that mass and
time do not appear in the dimensional formula for the wavelength. But they do appear in the
dimensional formulas for momentum and the Planck constant. So the final formula that we’re
looking for must combine ℎ and 𝑝 in such a way that the mass from momentum cancels
the mass from the Planck constant and time from momentum cancels the time from the
Planck constant, leaving overall dimensions of length.
Looking at our available formulas
then, we can immediately eliminate choice (B), ℎ times 𝑝 squared, and choice (E), ℎ
squared times 𝑝 squared, as possible answers. This is because we need to
eliminate mass and time from our final dimensional formula. But any time we multiply 𝑝 by ℎ,
we wind up with more factors of mass and time because mass times mass is mass
squared and time to the negative one times time to the negative one is time to the
negative two. So our final formula needs to
include division, not multiplication.
We can use a similar argument to
eliminate the choice (C), ℎ divided by 𝑝 squared. The dimensions of 𝑝 squared are 𝑀
squared times 𝐿 squared times 𝑇 to the negative two, just doubling all the
exponents in the dimensional formula for 𝑝. But as we can see, 𝑝 squared
actually has one more factor of mass and time than ℎ does. So dividing ℎ by 𝑝 squared still
leaves us with overall dimensions that include mass and time. This leaves us with 𝑝 divided by ℎ
and ℎ divided by 𝑝. Since these two are reciprocals of
one another, once we find the dimensions of one of them, we’ll know the dimensions
of the other just by changing the sign on all of the exponents.
So let’s find the dimensions of 𝑝
over ℎ. The dimensions of 𝑝 over ℎ are
given by 𝑀 times 𝐿 times 𝑇 to the negative one times 𝑀 to the negative one times
𝐿 to the negative two times 𝑇. The first three dimensions are from
𝑝, and the second three are from ℎ, with the sign reversed on each exponent. Because dividing by something is
the same as multiplying by that thing raised to the negative one power.
When we simplify this formula, we
note that 𝑀 times 𝑀 to the negative one power is dimensionless, as is 𝑇 to the
negative one power times 𝑇. 𝐿 times 𝐿 to the negative two is
equal to 𝐿 to the negative two plus one, which is just 𝐿 to the negative one. So the dimensions of 𝑝 over ℎ are
𝐿 to the negative one power.
Now, this is not what we need
because we’re looking for dimensions of 𝐿. However, it is very close. It’s the reciprocal of what we
need. This means that ℎ over 𝑝, the
reciprocal of 𝑝 over ℎ, will have dimensions of 𝐿, which is exactly what we’re
looking for. Therefore, we know that (D) is the
correct answer.
Although, as we’ve shown, we can
derive the correct answer using dimensional analysis, the de Broglie wavelength
relation is so important and so simple that it is worth just memorizing and knowing
the answer without having to do any work.
