Problem 55 - University Physics with Modern Physics

Question

A hydrogen atom in an \(n=2, l=1, m_{l}=-1\) state emits a photon when it decays to an \(n=1, l=0, m_{l}=0\) ground state. (a) In the absence of an external magnetic field, what is the wave-length of this photon? (b) If the atom is in a magnetic field in the \(+z\) -direction and with a magnitude of 2.20 \(\mathrm{T}\) , what is the shift in the wavelength of the photon from the zero-field value? Docs the magnetic field increase or decrease the wavelength? Disregard the effect of electron spin. [Hint: Use the result of Problem \(39.56(\mathrm{c}) . ]\)

Step-by-Step Solution

Answer

The wavelength of the photon is 122 nm. The magnetic field decreases the wavelength by about \(-4.82\times10^{-11}\) m.

Step 1: Determine the energy levels of the initial and final states

Using the formula for the energy levels of a hydrogen atom, \( E_n = -\frac{13.6eV}{n^2} \), calculate the energy levels for \( n=2 \) and \( n=1 \). For \( n=2 \), \( E_2 = -\frac{13.6eV}{2^2} = -3.4 \text{ eV} \). For \( n=1 \), \( E_1 = -\frac{13.6eV}{1^2} = -13.6 \text{ eV} \).

Step 2: Calculate the energy of the emitted photon

The energy of the emitted photon is equal to the difference in energy between the initial and final states. This is calculated as \( \Delta E = E_1 - E_2 = -13.6\text{ eV} + 3.4\text{ eV} = 10.2\text{ eV} \).

Step 3: Determine the wavelength of the photon

The energy of a photon is related to its wavelength by the equation \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant \( (6.626 \times 10^{-34} \text{Js}) \), and \( c \) is the speed of light \( (3.00 \times 10^8 \text{m/s}) \). Rearrange the formula to solve for the wavelength \( \lambda = \frac{hc}{E} \). Convert the energy to joules: \( 10.2 \text{eV} = 10.2 \times 1.602 \times 10^{-19} \text{J} \), so \( \lambda = \frac{(6.626 \times 10^{-34} \text{Js}) (3.00 \times 10^8 \text{m/s})}{10.2 \times 1.602 \times 10^{-19} \text{J}} \approx 1.22 \times 10^{-7} \text{m} \) or 122 nm.

Step 4: Calculate the Zeeman shift due to the magnetic field

The formula for the Zeeman energy shift is \( \Delta E = \mu_B B m_l \), where \( \mu_B \) is the Bohr magneton (\(9.274 \times 10^{-24} \text{J/T}\)), \( B \) is the magnetic field, and \( m_l \) is the magnetic quantum number. Here, \( m_l = -1 \) and \( B = 2.20 \text{T} \). The shift is \( \Delta E = 9.274 \times 10^{-24} \text{J/T} \times 2.20 \text{T} \times (-1) = -2.04 \times 10^{-23} \text{ J} \).

Step 5: Calculate the wavelength shift due to the Zeeman effect

The new energy of the photon is \( E' = \Delta E + 10.2 \text{eV} = 10.2 \times 1.602 \times 10^{-19} \text{J} - 2.04 \times 10^{-23} \text{J} \). The new wavelength is found using \( \lambda' = \frac{hc}{E'} \). The shift in the wavelength \( \Delta \lambda = \lambda' - \lambda \approx -4.82 \times 10^{-11} \text{m} \), indicating that the wavelength decreases.

Step 6: Conclusion: Determine the effect of the magnetic field on the wavelength

Since the shift \( \Delta \lambda \) is negative, the magnetic field causes the wavelength of the emitted photon to decrease.

Key Concepts

Energy Levels

In quantum mechanics, atoms like hydrogen have specific energy levels dictated by the principal quantum number, \( n \). For a hydrogen atom, the energy of an electron in a given level \( n \) is given by the formula:

• \( E_n = -\frac{13.6 \, \text{eV}}{n^2} \),

where 13.6 eV is the ionization energy of hydrogen. The negative sign indicates these are bound states. The energy levels determine the potential transitions an electron can make between different states. For instance, an electron can move from a higher energy level \( (n=2) \) to a lower one \( (n=1) \) by releasing energy in the form of a photon.
This transition between states has critical implications. When an electron moves from \( n=2 \) to \( n=1 \), the energy difference is 10.2 eV, calculated as:

• \( \Delta E = E_{1} - E_{2} = -13.6 \, \text{eV} + 3.4 \, \text{eV} \).

Photon Emission

Photons are packets of light energy. When an electron moves from a higher energy level to a lower one in an atom, a photon is emitted. The energy of the photon corresponds to the difference in energy between the initial and final states of the electron.
The energy of a photon \( E \) is related to its wavelength \( \lambda \) by the equation:

• \( E = \frac{hc}{\lambda} \),

where \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{Js} \)), and \( c \) is the speed of light (\( 3.00 \times 10^{8} \, \text{m/s} \)).
In the example of a hydrogen atom's transition from \( n=2 \) to \( n=1 \), a photon with energy 10.2 eV is emitted, and by converting this to a wavelength, we found that the photon has a wavelength of roughly 122 nm, which is in the ultraviolet range.

Zeeman Effect

The Zeeman Effect describes the influence of an external magnetic field on the energy levels of atomic electrons. This effect results in shifts in the spectral lines of atoms subjected to a strong magnetic field.

• The energy shift \( \Delta E \) is given by: \( \Delta E = \mu_B B m_l \),

where \( \mu_B \) is the Bohr magneton, \( B \) is the magnetic field strength, and \( m_l \) is the magnetic quantum number indicating the orientation of the electron's angular momentum.
In the exercise, with a magnetic field of 2.20 Tesla directed in the positive z-axis, the energy shift is negative, thus affecting the wavelength of the emitted photon. This results in a slight shift in the wavelength of the photon. Such shifts help scientists gain insights into both atomic structure and external magnetic fields.

Magnetic Quantum Number

The magnetic quantum number, denoted as \( m_l \), specifies the orientation of the electron's orbital around the nucleus in a magnetic field. It ranges from \( -l \) to \( +l \), where \( l \) is the azimuthal quantum number (related to angular momentum).
In a magnetic field, the value of \( m_l \) determines the additional energy experienced by the electron due to the Zeeman Effect. A change in \( m_l \) shifts the energy level, as seen in the example where \( m_l = -1 \).
This subtle change in energy alters how an electron behaves in the presence of a magnetic field, affecting the emission spectrum of photons. The magnetic quantum number is crucial in understanding the atom's behavior under various external conditions and plays a significant role in the Zeeman Effect observed in the spectral lines.