### Video Transcript

The wavelength ๐ of an electron
that has a kinetic energy of ๐ธ can be expressed as ๐ equals ๐ times โ divided by
๐ธ, where โ is the Planck constant and ๐ is a variable. Which of the following is equal to
๐? (A) Half the velocity of the
electron. (B) Half the momentum of the
electron. (C) Half the mass of the
electron. (D) None of the answers are
correct.

Weโre considering here this
equation, where ๐ is the wavelength of an electron with kinetic energy ๐ธ and โ is
the Planck constant. We want to figure out which one of
our answer options describes this variable ๐. One way to eliminate some of these
options is to think about the units on either side of our equation. Because this is an equation, the
units on the left must match those on the right.

If we work in SI base units where
possible, then the units of wavelength will be meters, the units of the Planck
constant โ will be joules times seconds, and the units of the kinetic energy of our
electron will be joules. We can therefore write this
equation, which is just an equation of units. On the left, we have meters, the
units of wavelength, and then we have the units of ๐, whatever those are,
multiplied by joules times seconds divided by joules.

We can see that the units of joules
in numerator and denominator will cancel out. And this means that the units of
this variable ๐ multiplied by seconds must be equal to meters. This equation then requires that
the units of ๐ are meters per second. That way this factor of seconds in
the numerator will cancel with this factor in the denominator. And weโll arrive at a simplified
equation, which says that meters is equal to meters, which is true. Since the units of ๐ are some
distance, in this case in units of meters, divided by some time, in this case in
units of seconds, we know that ๐ must represent a speed or a velocity.

Looking at our answer options, we
know that option (B) is not correct. ๐ canโt be a momentum because the
units of momentum are not the same as those of velocity. For the same reason, option (C)
canโt be correct either. The units of mass are not the same
as those of velocity. At this point, weโre inclined
towards answer option (A). Notice though that this option has
this somewhat strange phrasing โHalf the velocity of the electron.โ So far, weโve identified that ๐ is
a velocity, but we havenโt yet found that itโs half of anything. To see whether thatโs the case or
not, weโll need to approach this question from a different perspective.

Recall that weโre considering an
electron that has a wavelength ๐. That wavelength is described by the
de Broglie equation. And this equation says that the
wavelength of some particle is equal to Planckโs constant โ divided by the
particleโs momentum ๐. The momentum ๐ of an object is
equal to its mass times its velocity. We have then that the de Broglie
wavelength of some object equals Planckโs constant divided by the objectโs mass
times its velocity.

Clearing some space to work, we can
apply the de Broglie wavelength of the electron to this given equation. That is, in addition to saying that
๐ is equal to ๐ times โ divided by ๐ธ, we can say thatโs equal to โ divided by ๐
times ๐ฃ. Focusing on just this equation, we
can see that the Planck constant โ is common to both sides, and therefore we can
cancel it out.

We have then that our variable ๐
divided by the kinetic energy of the electron equals one divided by the electronโs
mass times its velocity. Letโs now recall that an objectโs
kinetic energy โ weโll refer to it as ๐ธ โ is equal to one-half that objectโs mass
times its velocity squared. Making this substitution, we find
that ๐ divided by one-half ๐๐ฃ squared equals one over ๐ times ๐ฃ.

If we multiply both sides of this
equation by ๐ times ๐ฃ, then on the left-hand side, one factor of ๐ and one factor
of ๐ฃ cancel from numerator and denominator. And on the right-hand side, both of
these factors cancel out completely. Our equation simplifies to ๐
divided by one-half times ๐ฃ is equal to one. Multiplying both sides by one-half
times the velocity ๐ฃ, that factor cancels on the left, and we find the final
simplified result that ๐ equals ๐ฃ divided by two.

This then is what answer option (A)
is speaking to when it describes ๐ as half the velocity of the electron. ๐ is literally equal to ๐ฃ divided
by two. We choose then answer option
(A). The variable ๐ is equal to half
the velocity of the electron.