Question

The momentum of a photon is $p$, the corresponding wavelength is (1) $h/p$ (2) $hp$ (3) $p/h$ (4) $h/\sqrt{p}$

Short answer

Option (1) ${\displaystyle \frac{h}{p}}$

Step-by-step solution

Understand the Relationship

Recall that the momentum ($p$) of a photon is related to its wavelength ($\lambda$) by the equation $p={\displaystyle \frac{h}{\lambda }}$ where $h$ is Planck's constant.

Solve for Wavelength

Rearranging the equation $p={\displaystyle \frac{h}{\lambda }}$ to solve for $\lambda$, we get: $$\lambda ={\displaystyle \frac{h}{p}}$$

Match with Given Choices

Compare the derived formula $\lambda ={\displaystyle \frac{h}{p}}$ with the given options. Option (1) ${\displaystyle \frac{h}{p}}$ matches our derived formula.

Conclusion

The corresponding wavelength $\lambda$ is given by option (1) ${\displaystyle \frac{h}{p}}$.

Key concepts

Wavelength Calculation

The calculation of wavelength for a photon is a fundamental concept in physics. A photon's wavelength is inversely proportional to its momentum. This relationship is given by the formula: $$p={\displaystyle \frac{h}{\lambda }}$$ Here, $p$ represents the photon's momentum, $\lambda$ denotes the wavelength, and $h$ is Planck's constant. Understanding this formula allows us to calculate the wavelength of a photon if we know its momentum. By rearranging the formula, we can solve for the wavelength: $$\lambda ={\displaystyle \frac{h}{p}}$$ This means that the wavelength is equal to Planck's constant divided by the momentum. Using this equation correctly can help solve a variety of physics problems related to photon behavior.

Planck's Constant

Planck's constant is a vital constant in physics, symbolized by $h$. It has a value of approximately $6.626\times {10}^{-34}$ joule-seconds (Js). Planck's constant plays an essential role in quantum mechanics. It describes the sizes of quanta, the smallest units possible in energy transitions. Without this constant, many quantum mechanical equations would not work. In the context of photon's momentum and wavelength, Planck's constant provides that bridge or proportional relationship. It shows that despite photons lacking mass, they indeed carry momentum, which can be calculated using $p={\displaystyle \frac{h}{\lambda }}$. Applications of Planck's constant span various domains from defining the scales of quantum effects to aiding in technologies like quantum computing and understanding atomic spectra.

Physics Formulas

Physics is full of formulas that describe how different physical properties relate to one another. When dealing with photons, key formulas help explain their behavior. One critical formula is $E=hu$, which describes the energy $E$ of a photon as the product of Planck's constant $h$ and its frequency $u$. Another crucial equation, as seen in our exercise, is $p={\displaystyle \frac{h}{\lambda }}$, linking a photon's momentum to its wavelength. Here, $\lambda$ represents the wavelength, and Planck's constant $h$ ensures the equation's dimensional correctness. These formulas are used not just in theoretical physics but also in practical applications, from calculating the energies of particles to designing electronic devices like LEDs and understanding behaviors of waves in various contexts. Knowing and understanding these key formulas empowers students to tackle a diverse range of physics problems with confidence.