Okay, let's try to tackle this question step by step. So, the question is about comparing the order of magnitude of the paramagnetic coupling term (⟨H⟩) with the transition energy ΔE of a hydrogen atom when it's subjected to a magnetic field of 1 Tesla. The options given are A to D, each indicating whether ⟨H⟩ is much less than, greater than, equal to, or much greater than ΔE.

First, I need to recall what the paramagnetic coupling term is. From what I remember, paramagnetic coupling arises due to the Zeeman effect, which is the splitting of atomic energy levels when placed in a magnetic field. The Zeeman effect has two main contributions: the orbital Zeeman effect and the spin Zeeman effect. Since the question mentions the magnetic quantum number m (m being small), I think it's referring to the orbital Zeeman effect because the spin quantum number is usually denoted by s and m_s. Wait, but the problem says "m" is the orbital magnetic quantum number. So, yes, we're dealing with the orbital part here.

The paramagnetic coupling term, which is ⟨H⟩ here, would be proportional to the magnetic field strength and the magnetic quantum number m. The formula for the Zeeman effect for the orbital part is given by:

⟨H⟩ = \frac{\mu_B B m}{\hbar}

Wait, actually, the Zeeman term is usually written as E_Zeeman = -μ·B, where μ is the magnetic moment. The magnetic moment for orbital motion is μ = (eħ)/(2m_e) * m, right? Because the magnetic moment is proportional to the angular momentum, which is m * ħ. So, μ = (eħ/(2m_e)) * m. Then the energy would be E = -μ·B. But since the magnetic field is along the z-axis (Oz), the energy becomes E = -μ_z B. The magnitude would be |E| = μ B / something. Wait, maybe I should think in terms of the actual formula.

The energy shift due to the Zeeman effect is given by ΔE = μ_B * m * B, where μ_B is the Bohr magneton. Wait, but wait, the orbital magnetic moment is μ = (eħ/(2m_e)) * m. The Bohr magneton is μ_B = (eħ)/(2m_e). So, the energy shift is E_Zeeman = μ * B = (eħ/(2m_e)) * m * B. Which is μ_B * m * B. So, ⟨H⟩ here is the expectation value of the Hamiltonian due to the magnetic field, which would be this E_Zeeman. So, ⟨H⟩ = μ_B * m * B.

But the problem states that m (the orbital magnetic quantum number) has small values. For hydrogen, the possible values of m are integers from -l to +l, where l is the azimuthal quantum number. For a given energy level (n), l can be from 0 to n-1. But since m is small, maybe we're considering states where l is small, like l=0 (s orbital), but m=0 in that case. Wait, but if m is small, maybe l is not zero. For example, if l=1, then m can be -1, 0, +1. So m is small in magnitude, but l can be larger. But regardless, the key is that m is small compared to what? The problem says "small values of m", which might mean that m is much less than 1, but that doesn't make sense because m is an integer. Wait, maybe it's referring to the magnitude being small compared to the typical quantum numbers in hydrogen. Hmm, perhaps the point is that m is small compared to the principal quantum number n or something else. But maybe the actual value of m is small, like m=1 or m=0, since for higher l, m can be larger. But regardless, the question is about the order of magnitude between ⟨H⟩ and ΔE.

Now, the transition energy ΔE is given by the energy difference between two states in hydrogen. The problem states that the wavelength λ is 0.4861 μm. Let's compute the energy corresponding to this wavelength.

The energy of a photon is E = hc/λ, where h is Planck's constant, c is the speed of light, and λ is the wavelength. Let's compute that.

First, convert λ from micrometers to meters. 0.4861 μm = 0.4861 x 10^-6 meters.

h = 6.626 x 10^-34 J·s

c = 3 x 10^8 m/s

So E = (6.626e-34 * 3e8) / (0.4861e-6)

Calculate numerator: 6.626e-34 * 3e8 = 1.9878e-25 J·m

Divide by 0.4861e-6 m: 1.9878e-25 / 0.4861e-6 = approx 4.096e-19 J

Convert Joules to electronvolts (eV) since we're comparing with typical magnetic energies. 1 eV = 1.602e-19 J, so divide by 1.602e-19:

4.096e-19 / 1.602e-19 ≈ 2.56 eV

So ΔE ≈ 2.56 eV.

Now, ⟨H⟩ is the Zeeman energy term. Let's compute that. The formula is ⟨H⟩ = μ_B * m * B, where μ_B is the Bohr magneton, which is approximately 9.274e-24 J/T.

Given B = 1 T, and m is small. Let's consider possible values of m. Since m is the orbital magnetic quantum number, for hydrogen with a given l, m can be -l, ..., +l. For example, if the electron is in an s orbital (l=0), then m=0. If it's in p (l=1), m can be -1, 0, +1. If it's in d (l=2), m can be -2, -1, 0, +1, +2, etc. The question states that m is small, so maybe m is 0, but if m=0, then ⟨H⟩ would be zero. But perhaps the electron is in a p orbital with m=±1, which is still small compared to l=1. But maybe the question is considering that m is small compared to some other scale, but in any case, let's proceed.

If m is 1, then ⟨H⟩ = μ_B * 1 * 1 T = 9.274e-24 J. Let's compare that to ΔE which is 4.096e-19 J. So 9.274e-24 J is much smaller than 4.096e-19 J. So ⟨H⟩ << ΔE. But the question says "small values of m", so maybe m is even smaller. For example, if m=1 (which is the maximum for l=1), but even so, the Zeeman energy is 1e-5 times the transition energy. So ⟨H⟩ is much smaller than ΔE. Therefore, the order of magnitude would be ⟨H⟩ << ΔE, which is option A.

Wait, but let's check again. Let me compute the actual numbers more precisely.

Compute ΔE:

E = hc/λ = (6.62607015e-34 J·s * 299792458 m/s) / 0.4861e-6 m

But using approximate values:

h ≈ 6.626e-34 J·s

c ≈ 2.998e8 m/s

λ ≈ 0.4861e-6 m

So E = (6.626e-34 * 2.998e8) / 0.4861e-6

Calculate numerator: 6.626e-34 * 2.998e8 = (6.626 * 2.998) e-34+8 = approx 19.86 e-26 J·m

Divide by 0.4861e-6 m: 19.86e-26 / 0.4861e-6 ≈ (19.86 / 0.4861) e-26+6 ≈ 40.93e-20 = 4.093e-19 J

Convert to eV: 4.093e-19 J / 1.602e-19 J/eV ≈ 2.556 eV

So ΔE ≈ 2.556 eV.

Now, ⟨H⟩ = μ_B * m * B. μ_B = 9.274e-24 J/T.

Assuming m=1 (since m is small, but the maximum possible for a given l):

⟨H⟩ = 9.274e-24 J/T * 1 * 1 T = 9.274e-24 J

Convert to eV: 9.274e-24 / 1.602e-19 ≈ 5.79e-5 eV

So ⟨H⟩ ≈ 5.79e-5 eV, which is much smaller than ΔE ≈ 2.556 eV. Therefore, ⟨H⟩ << ΔE, so option A is correct.

But wait, the problem states that the magnetic field is 1 T. What if the electron's spin is considered? Wait, no, the problem mentions the magnetic quantum number m, which is for orbital angular momentum. The spin magnetic quantum number is m_s, which isn't mentioned here. So we're only dealing with the orbital Zeeman effect.

Another point: the question says "the order of magnitude of the paramagnetic coupling term with transition energy". So we need to compare ⟨H⟩ and ΔE. Since ⟨H⟩ is 5.79e-5 eV and ΔE is 2.556 eV, the ratio is about 2.2e-5. So ⟨H⟩ is about 5.7e-5 eV, which is 5.7e-5 times ΔE. So ⟨H⟩ is much smaller than ΔE, so option A is correct.

But let's check if there's any possible m value that could make ⟨H⟩ comparable to ΔE. For example, if m is 100, but that's impossible because m can't exceed l, and l is at most n-1. However, for higher energy levels (higher n), l can be larger. But the problem states that m is small, so even if n is large, m is small. For example, if l is 100, then m can be up to 100, but m is still considered small compared to l. Wait, but if m is 100 and l is 100, then ⟨H⟩ = μ_B * 100 * 1 T = 9.274e-24 * 100 = 9.274e-22 J. Convert to eV: 9.274e-22 / 1.602e-19 ≈ 5.79e-3 eV, which is still much smaller than 2.556 eV. So even with m=100, ⟨H⟩ is 5.79e-3 eV, which is about 0.00225 times ΔE. So still ⟨H⟩ << ΔE.

Wait, but even if m is 1000, but that's not a realistic value for hydrogen. For hydrogen, the principal quantum number n must be at least l+1, so if l=1000, n would be 1001. But the problem says m is small, so maybe m=1. Therefore, regardless of the actual value, the conclusion is that ⟨H⟩ is much smaller than ΔE.

Therefore, the correct answer should be option A: ⟨H⟩ << ΔE.
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