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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

Respuesta :

Answer:

Part a)

[tex]\lambda = 0.345 m[/tex]

Part b)

[tex]\Delta x = 0.274 m[/tex]

Part c)

[tex]r = 2.8 \times 10^{11} m[/tex]

Explanation:

Part a)

De broglie wavelength is given as

[tex]\lambda = \frac{h}{mv}[/tex]

[tex]\lambda = \frac{1}{(0.145)(20)}[/tex]

[tex]\lambda = 0.345 m[/tex]

Part b)

By principle of uncertainty we know that

[tex]\Delta x \times \Delta P = \frac{h}{4\pi}[/tex]

[tex]\Delta x \times (0.145)(21 - 19) = \frac{1}{4\pi}[/tex]

[tex]\Delta x = 0.274 m[/tex]

Part c)

As we know that

[tex]\frac{kq_1q_2}{r^2} = \frac{mv^2}{r}[/tex]

also we know

[tex]mvr = \frac{nh}{2\pi}[/tex]

[tex]v = \frac{h}{2\pi mr}[/tex]

now we have

[tex]\frac{ke^2}{r} = \frac{mh^2}{4\pi^2m^2 r^2}[/tex]

[tex]r = \frac{h^2}{4\pi^2mke^2}[/tex]

[tex]r = 2.8 \times 10^{11} m[/tex]

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