Question

The de-Broglie wavelength of a tennis ball of mass 60 $g$ moving with a velocity of 10 $m/s$ is, approximately (planck's constant, $h = 6.63 \times {10^{ - 34}}J - s$)
A. ${10^{ - 33}}m$
B. ${10^{ - 31}}m$
C. ${10^{ - 16}}m$
D. ${10^{ - 25}}m$

Answer 1

Hint: According to the quantum mechanical model of the atom two major ideas contribute toward the atomic existence. The first being the dual behavior of the atom and the second being the Heisenberg principle of uncertainty. Both of these ideas contribute to the quantum mechanical model. The dual nature of the atom consists of the particle nature and the wave nature.

Complete step by step answer:
> The quantum mechanical model states the existence of two ideas that operate in the atom, dual behavior of matter, and the Heisenberg uncertainty principle.
> According to de Broglie the matter, like radiation should show dual behavior means it should contain the properties of a wave and that of a particle both.
This means that an atom will have both momentum and wavelength.
From this analogy, de Broglie gave the relationship between the wavelength ( $\lambda $ ) and momentum ( $p$ ) of the matter which can be expressed as,
$\lambda = \dfrac{h}{p}$
Since momentum $p$ can be expressed as $p = mv$ , where m is the mass of the object and v is the velocity of the object, the above relationship can be simplified as
$\lambda = \dfrac{h}{{mv}}$ , where h is called Planck’s constant.
Considering the case above, the mass of the object is given to be 60 $g$ $ = 6 \times {10^{ - 2}}kg$ and the velocity of the object is 10 $m/s$ .
With the help of the above relationship we can find the wavelength as
$\lambda = \dfrac{h}{{mv}}$
$ \Rightarrow \lambda = \dfrac{{6.63 \times {{10}^{ - 34}}}}{{6 \times {{10}^{ - 2}} \times 10}}$
$ \Rightarrow \lambda = 1.105 \times {10^{ - 33}}m$

So, the correct answer is Option A.

Note: 

1. The wavelengths associated with ordinary objects are very short to be detected by us while the wavelengths of subatomic particles can be experimentally detected.
2. The Heisenberg uncertainty principle states that it is impossible to determine simultaneously, the exact position and the exact momentum (or velocity) of an electron.
Mathematically it can be given as,
$\Delta x\Delta p \geqslant \dfrac{h}{{2\pi }}$