Question

If a hydrogen atom emits a Photon of wave length \(\lambda\). the recoil speed of the atom of mass \(\mathrm{m}\) is given by (A) \((\mathrm{h} / \mathrm{m} \lambda)\) (B) \((\mathrm{mh} / \lambda)\) (C) \(\operatorname{mh} \lambda\) (D) \((\mathrm{m} \lambda / \mathrm{h})\)

Short answer

The short answer is: The recoil speed of the hydrogen atom is given by v = \(\frac{h}{m \lambda}\), which matches option (A).

Step-by-step solution

Understand the given parameters and variables

We are given the following variables and parameters: - Wavelength of the emitted photon: \(\lambda\) - Mass of the hydrogen atom: m - Planck's constant: h Our task is to find the correct expression for the recoil speed of the hydrogen atom, given these parameters.

Find the momentum of the emitted photon

According to the de Broglie hypothesis, the momentum of a photon can be expressed as: Momentum of the photon (p_photon) = \(\frac{h}{\lambda}\) Where h is the Planck's constant and \(\lambda\) is the wavelength of the emitted photon.

Apply the conservation of momentum to find the momentum of the hydrogen atom

According to conservation of momentum, the momentum of the hydrogen atom after the emission of the photon is equal and opposite to the momentum of the emitted photon. Thus, the momentum of the hydrogen atom can be expressed as: Momentum of the hydrogen atom (p_hydrogen) = -p_photon

Find the recoil speed of the hydrogen atom

The momentum of the hydrogen atom can be related to its mass, m, and its recoil speed, v, by the following expression: p_hydrogen = m * v We can now substitute the expression for p_hydrogen from Step 3: m * v = -\(\frac{h}{\lambda}\) Now, we need to find the expression for the recoil speed, v. We can do this by isolating v: v = -\(\frac{h}{m \lambda}\) Considering that only the magnitudes are shown in the options, we can disregard the negative sign: v = \(\frac{h}{m \lambda}\) Comparing this expression to the provided options, we can see that it matches option (A). Thus, the recoil speed of the hydrogen atom is given by: v = \(\frac{h}{m \lambda}\)

Key concepts

Photon Emission

In the world of atoms and particles, **photon emission** is an intriguing phenomenon. When a hydrogen atom transitions from a higher energy state to a lower energy state, a photon is emitted. Photons are particles of light, and this emission is a way for the atom to shed excess energy.

One key aspect of photon emission is that it occurs at specific wavelengths, which are related to the energy difference between the initial and final states of the atom. These emissions form what is known as an atomic spectrum. The emitted photon carries away the energy, and this is what attributes to the characteristic spectral lines often observed. Each element has its own unique set of lines, acting like its fingerprint.

Additionally, the emission process results in the atom experiencing a recoil, much like a shooting rifle recoils when a bullet is fired. This recoil occurs because the emitted photon has momentum, a concept essential to understanding phenomena on such a small scale.

De Broglie Hypothesis

The **de Broglie Hypothesis** is a foundational concept in quantum mechanics, offering a bridge between particle and wave phenomena. Proposed by Louis de Broglie in the early 20th century, it introduced the revolutionary idea that particles of matter, like electrons and atoms, exhibit wave-like properties.

According to de Broglie, the wavelength (\(\lambda\)) of any particle is inversely proportional to its momentum (\(p\)). This is expressed through the formula:\(\lambda = \frac{h}{p}\), where \(h\) is Planck's constant. For photons, the momentum \(p_photon\) is given by:\[\frac{h}{\lambda}\].

This wave-particle duality lays down the foundation for many quantum mechanics principles, such as the behavior of electrons in atoms and the way particles like photons exhibit both wave-like and bullet-like behavior. It beautifully demonstrates how nature's smallest entities do not conform strictly to the categories we perceive in everyday life.

Conservation of Momentum

The principle of **conservation of momentum** is a cornerstone of classical and quantum physics. It asserts that in an isolated system, the total momentum remains constant, regardless of the interactions occurring within.

When a photon is emitted by an atom, the conservation of momentum comes into play. Before the photon is emitted, the total momentum of the system is just the atom's momentum since the photon isn't yet present. After emission, both the atom and the photon have momentum. To ensure the total momentum stays the same, the atom must recoil. It gains a velocity that results in momentum equal in magnitude but opposite in direction to that of the photon. This relationship is what ultimately determines the recoil speed of an atom when a photon is emitted, as seen in the provided exercise solution.

This principle helps explain numerous phenomena across physics, from everyday occurrences like a bat hitting a ball to the interactions taking place at a subatomic level, emphasizing the natural inclination of systems to maintain their "momentum balance."

Planck's Constant

**Planck's Constant** is a fundamental constant that plays a vital role in quantum mechanics. Denoted as \(h\), it has a value of approximately \(6.626 \times 10^{-34} \text{ Js}\). Max Planck introduced this constant in the early 20th century when he formulated quantum theory to explain black-body radiation.

Planck's constant is essential in the quantization of energy, where it serves as a proportionality constant linking the energy of a photon to its frequency:\(E = hf\), where \(E\) is energy and \(f\) is frequency. Through the de Broglie hypothesis, it is also used to connect the momentum and wavelength of particles, underscoring its versatile role across various physical principles.

In the photon emission process discussed in the exercise, Planck's constant helps determine both the momentum of the emitted photon and consequently, the recoil speed of the emitting atom. Thus, Planck's constant is a bridge between macroscopic observations and quantum phenomena, delivering profound insights into how the universe operates at its most fundamental level.