Question

It is interesting to speculate on the properties of a universe with different values for the fundamental constants.

a. In a universe in which Planck’s constant had the value h = 1 J s, what would be the de Broglie wavelength of a 145 g baseball moving at a speed of 20 m/s?
b. Suppose the velocity of the ball from part (a) is known to lie between 19 and 21 m/s. What is the smallest distance within which it can be know to lie?
c. Suppose that in this universe the mass of the electron is 1 g and the charge on the electron is 1 C. Calculate the Bohr radius of the hydrogen atom in this universe

Answer 1

Part a)

`\lambda = 0.345 m`

Part b)

`\Delta x = 0.274 m`

Part c)

`r = 2.8 * 10^(11) m`

Step-by-step explanation:

Part a)

De broglie wavelength is given as

`\lambda = (h)/(mv)`

`\lambda = (1)/((0.145)(20))`

`\lambda = 0.345 m`

Part b)

By principle of uncertainty we know that

`\Delta x * \Delta P = (h)/(4\pi)`

`\Delta x * (0.145)(21 - 19) = (1)/(4\pi)`

`\Delta x = 0.274 m`

Part c)

As we know that

`(kq_1q_2)/(r^2) = (mv^2)/(r)`

also we know

`mvr = (nh)/(2\pi)`

`v = (h)/(2\pi mr)`

now we have

`(ke^2)/(r) = (mh^2)/(4\pi^2m^2 r^2)`

`r = (h^2)/(4\pi^2mke^2)`

`r = 2.8 * 10^(11) m`