Find the uncertainty in velocity, if the uncertainty in position is equal to the uncertainty in momentum.
a) [tex]$\frac{h}{\sqrt[2]{\pi m}}$[/tex]
b) [tex]$\frac{1}{2 m} \sqrt{\frac{h}{\pi}}$[/tex]
c) [tex]$\frac{1}{m} \sqrt{\frac{h}{\pi}}$[/tex]
d) [tex]$\frac{1}{2} \sqrt{\frac{h}{m \pi}}$[/tex]
Answer (2)
Use Heisenberg's uncertainty principle: Δ x Δ p ≥ 4 π h .
Apply the condition Δ x = Δ p to get ( Δ x ) 2 ≥ 4 π h , which implies Δ x ≥ 4 π h .
Use the relation Δ p = m Δ v and Δ x = Δ p to get m Δ v ≥ 4 π h .
Solve for Δ v to find the uncertainty in velocity: Δ v = 2 m 1 π h .
The final answer is 2 m 1 π h .
Explanation
Problem Analysis We are given that the uncertainty in position, Δ x , is equal to the uncertainty in momentum, Δ p . Our goal is to find the uncertainty in velocity, Δ v . We will use Heisenberg's uncertainty principle to relate these quantities.
Heisenberg's Uncertainty Principle Heisenberg's uncertainty principle states that the product of the uncertainty in position and the uncertainty in momentum is greater than or equal to a constant: Δ x Δ p ≥ 4 π h where h is Planck's constant.
Applying the Given Condition Since we are given that Δ x = Δ p , we can substitute Δ x for Δ p in the uncertainty principle: Δ x ⋅ Δ x ≥ 4 π h This simplifies to: ( Δ x ) 2 ≥ 4 π h Taking the square root of both sides, we get: Δ x ≥ 4 π h
Relating Momentum and Velocity We know that momentum is related to velocity by the equation p = m v , where m is the mass and v is the velocity. Therefore, the uncertainty in momentum can be written as Δ p = m Δ v . Since Δ x = Δ p , we can write: Δ x = m Δ v
Solving for Uncertainty in Velocity Now we can substitute the expression for Δ x from step 3 into the equation from step 4: m Δ v ≥ 4 π h Solving for Δ v , we get: Δ v ≥ m 1 4 π h
Simplifying the Expression We can simplify the expression for Δ v : Δ v ≥ m 1 4 π h = m 1 π h ⋅ 4 1 = m 1 π h ⋅ 2 1 = 2 m 1 π h Therefore, the uncertainty in velocity is: Δ v ≥ 2 m 1 π h
Final Answer The uncertainty in velocity is 2 m 1 π h . Therefore, the correct answer is option b) 2 m 1 π h .
Examples
Heisenberg's uncertainty principle, which we used to solve this problem, has practical applications in fields like quantum computing and microscopy. For example, in quantum computing, understanding the limits of how precisely we can know the position and momentum of a quantum particle helps in designing more accurate quantum algorithms. In microscopy, this principle affects the resolution we can achieve when imaging very small objects, as the more accurately we try to determine the position of a particle, the less accurately we know its momentum, and vice versa. This trade-off is fundamental in many areas of science and technology.
The uncertainty in velocity can be determined using Heisenberg's uncertainty principle. By establishing the relationship between uncertainty in position and momentum, we find that the uncertainty in velocity is given by Δ v ≥ 2 m 1 π h . Thus, the correct option is b) 2 m 1 π h .
;
