An alpha particle (mass =6.6 ×10^-24 g ) emitted by a radium isotope travels at 3.4 ×10^7 ±0.1

An alpha particle (mass =6.6 ×10^-24 g ) emitted by a...

An alpha particle (mass $=6.6 \times 10^{-24} \mathrm{~g}$ ) emitted by a radium isotope travels at $3.4 \times 10^{7} \pm 0.1 \times 10^{7} \mathrm{mi} / \mathrm{h} .$
(a) What is its de Broglie wavelength (in meters)?
(b) What is the uncertainty in its position?

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Key Concepts

Heisenberg Uncertainty Principle

This principle asserts that certain pairs of physical properties, such as position and momentum, cannot both be measured to arbitrary precision simultaneously. The principle is commonly presented as ?x ?p ? ?/2, indicating that an uncertainty in momentum leads to an inherent minimum uncertainty in the particle’s position, a key concept in quantum mechanics.

Momentum

Momentum, defined as the product of mass and velocity, is a crucial physical quantity in both classical and quantum physics. In the context of the de Broglie wavelength, it directly influences the value of the wavelength of a particle, thereby affecting its wave-like properties.

de Broglie Hypothesis

This concept states that particles can exhibit wave-like behavior, characterized by a wavelength inversely proportional to their momentum. The relationship, given by the formula ? = h/(mv), where h is Planck's constant, m is the mass, and v is the velocity, is fundamental in quantum mechanics for linking particle properties with wave properties.

Transcript

00:02 All right, so first we need to do some conversions because the question gave us units of miles per hour and grams when we don't want that.

00:13 So first we're going to convert our speed from miles per hour to meters per second.

00:24 So we know 3 .4 times 10 to 7 miles per hour.

00:30 So we want to convert it again into meters per second.

00:37 So you want to cancel the hour? so one hour is 3 ,600 seconds.

00:44 And one mile is 1609 .34 meters.

00:52 So our hours and our miles are going to cancel, and we're going to end up with meters per second.

01:00 So we end up getting 1 .52 times 10 to the same.

01:06 Meters per second.

01:10 So we also need to switch over our mass and grams to kilograms.

01:16 So 6 .6 times 10 to the minus 24 grams we want to divide by a thousand grams to get to get it in kilograms.

01:31 So grams cancel and it ends up being 6 .6 times 10 to the minus 27 kilograms.

01:40 So there is our speed and our mouse.

01:44 So now to find the number of a wavelength we know wavelength is h over mu and we know a h is planks constant so that's 6 .626 times 10 to the minus 34 kilograms meter squared over seconds and our mass we just found to be 6 .6 times 10 to the minus 27 kilograms and we're going to multiply that by 1 .52 times 10 to the 7th meters per second...

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