Final answer:
To find the initial quantum state for the hydrogen transition, we can use the Rydberg formula to calculate the initial quantum number based on the photon's emitted energy and the final quantum state. Once we have the energy difference, we solve for the initial quantum number squared and finally take the square root to get the initial quantum number.
Step-by-step explanation:
The student has asked how to determine the initial quantum state of hydrogen when the energy of an emitted photon is given. We know that the photon corresponds to a transition ending at the quantum state n = 2. Based on the Bohr model of the hydrogen atom, the energy emitted during an electron transition is equal to the energy difference between two energy levels. The Rydberg formula for the energy difference is given by:
E = R_H \left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right)
Where E is the energy of the emitted photon, R_H is the Rydberg constant (approximately 2.18 \times 10^{-18} J), n_f is the final quantum number, and n_i is the initial quantum number. The student has provided the energy of the emitted photon as 4.8430 \times 10^{-19} J, and we know the final quantum level n_f = 2 based on the Balmer series.
We can rearrange the formula to solve for the initial quantum number n_i:
\frac{1}{n_i^2} = \frac{1}{n_f^2} - \frac{E}{R_H}
Plugging in the given values:
\frac{1}{n_i^2} = \frac{1}{2^2} - \frac{4.8430 \times 10^{-19} \text{J}}{2.18 \times 10^{-18} \text{J}}
This calculation will give us the value of n_i^2, from which we can take the square root to find the initial quantum number n_i, which is the answer the student is looking for.