When a photon is emitted by an atom, the atom must recoil to...

When a photon is emitted by an atom, the atom must recoil to conserve momentum. This means that the photon and the recoiling atom share the transition energy. (a) For an atom with mass $m,$ calculate the correction $\Delta \lambda$ due to recoil to the wave length of an emitted photon. Let $\lambda$ be the wavelength of the photon if recoil is not taken into consideration. (Hint: The correction is very small, as Problem 39.56 suggests, so $|\Delta \lambda| / \lambda<<1 .$ Use this (b) Evaluate the correction for a hydrogen atom in which an electron in the $n$ th level returns to the ground level. How does the answer depend on $n$ ?

Key Concepts

Conservation of Momentum

When a photon is emitted from an atom, conservation of momentum requires that the atom recoils. This is a fundamental principle in physics stating that the total momentum before and after an event must remain constant. The recoil of the atom ensures that when the photon takes away momentum, the atom’s momentum adjusts in the opposite direction to satisfy this conservation law.

Conservation of Energy

During the emission process, the energy released from the atomic transition is shared between the energy of the photon and the kinetic energy associated with the recoil of the atom. This partitioning of energy is critical because the exact energy available to the photon is slightly less than the total transition energy, due to some energy being used to give the atom recoil motion.

Recoil Effect

The recoil effect refers to the kinetic energy acquired by the atom as it recoils following photon emission. Even though the atom is much more massive than the photon, a small fraction of the energy is diverted to the atom’s kinetic energy. This recoil energy causes a minute correction to the energy and consequently the wavelength of the emitted photon.

Perturbative Correction and Approximations

Since the recoil correction is typically very small relative to the photon’s energy, a perturbative or approximate correction method is used. In such cases, the change in the photon's wavelength can be expressed as a small correction term added to the wavelength calculated without considering recoil. This approach leverages the fact that the correction is much smaller than the main term, allowing simplifications in calculations.

Atomic Energy Transitions

The process of photon emission is intrinsically connected to atomic energy transitions, which occur when electrons move between quantized energy levels. The energy difference between these levels normally defines the energy, and hence the wavelength, of the photon. However, due to the recoil of the atom during emission, the precise wavelength is slightly modified, illustrating the subtle interplay between quantum and classical effects in atomic spectroscopy.

Transcript

00:01 In this question, we are basically looking at how when an atom emits a photon, there's always got to be some recoil of the atom or the nuclei because of conservation of momentum.

00:25 Right, at the start it is not moving.

00:31 Total momentum is zero.

00:33 After it emits a photon, the photon actually takes away some momentum, right? the photon has a momentum, therefore the nuclei must move in the opposite direction with the same magnitude of momentum, such that they will cancel each other out and will maintain a net momentum of zero.

01:03 So what you want to find is how much this recoil momentum or recoil energy affects the actual wavelength of photon that is being emitted.

01:22 Because this by conservation of energy, the energy that is in the recoil plus the photon will be equal to the transition energy.

01:35 Say it is transiting from an excited state down to the ground state.

01:43 It will emit the photon from say en to e1.

01:52 I'm going to call this the transition energy.

02:00 Just the energy of the transition.

02:06 So if you have to ignore, if we ignore the required energy, then the energy or transition will be fully the energy of our photon.

02:22 Right, and this will be equals to hc over lambda, where lambda is the, is the one that is given in the question.

02:32 This is the wavelength of the photon if recoil is not considered.

02:38 However, now we want to consider the recoil, so definitely the photon wavelength.

02:43 Length is different.

02:46 It's slightly different.

02:50 So how are we going to start from this question? right? we need to understand that the actual photon energy and actual photon wavelength.

03:05 Right? we're going to label it as this.

03:12 Therefore, if we...

03:14 This is also equals to energy of the transition, minus away, energy of the recoil.

03:21 This is the actual amount of energy that is being transferred to the photon.

03:36 Therefore, the actual wavelength of the photon is hc.

03:49 What we want is the correction delta lambda, which is actually goes to lambda prime minus so we take lambda prime because we expect this to be a slightly larger wavelength because of a lower energy.

04:16 So we expect this to be positive.

04:21 So we take lambda prime minus lambda where lambda is the wavelength assuming we are ignoring the recoil.

04:31 So it's just hc over energy of the transition...

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