The de Broglie wavelength of a particle that has kinetic energy K is λ. The wavelength λ is proportional to

Question

BlackTom AI · Accepted Answer

λ = h/p and for a nonrelativistic particle K = p^2/(2m), so these two relations determine the dependence of λ on K.

1. Relevant knowledge points and concepts
- de Broglie relation: λ = h/p, where h is Planck’s constant and p is momentum.
- Classical (nonrelativistic) kinetic energy: K = p^2/(2m), where m is the particle mass.
- Proportionality: when writing λ ∝ f(K), constants (h, m, numerical factors) do not affect the functional dependence on K.

2. Complete reasoning process step by step
Start with K = p^2/(2m). Solve for momentum p in terms of K:
p^2 = 2mK
p = sqrt(2mK). (We take the positive root for magnitude of momentum.)
Use de Broglie: λ = h/p
Substitute p: λ = h / sqrt(2mK).

3. Intermediate calculation steps shown
K = p^2/(2m) → p = (2mK)^(1/2)
λ = h / p = h / (2mK)^(1/2) = (h / sqrt(2m)) · K^(-1/2).

4. Why the correct answer is correct
The expression λ = (h / sqrt(2m)) · K^(-1/2) shows λ is proportional to K^(-1/2), i.e., inversely proportional to the square root of the kinetic energy. Thus λ ∝ 1/√K, which matches option b.

5. Why the other options are incorrect
- a (K^−2): implies λ ∝ 1/K^2, which is a much stronger dependence and inconsistent with the p ∝ √K relation.
- c (K^2): implies λ increases with the square of K; but λ actually decreases as K increases.
- d (1/K): implies simple inverse proportionality, but we have inverse square-root dependence, not inverse-linear.

6. Conclusion
Using λ = h/p and p = √(2mK) yields λ ∝ K^(−1/2). Therefore the correct choice is b (1/√K), which matches the provided answer.

The de Broglie wavelength of a particle that has kinetic energy K is λ. The wavelength λ is proportional to单项选择题

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