Calculate The Frequency From N 4 to N 3

When an electron transitions from the n=4 energy level to the n=3 energy level in a hydrogen-like atom, it emits a photon with a specific frequency. This calculator computes that frequency using quantum mechanics principles.

Introduction

In quantum mechanics, electrons in atoms occupy discrete energy levels (quantum states) described by the principal quantum number n. When an electron moves from a higher energy level (n=4) to a lower one (n=3), it emits a photon with energy equal to the difference between the two levels.

The frequency of this emitted photon is directly related to the energy difference between the levels. This phenomenon is fundamental to spectroscopy and understanding atomic structure.

Formula

The frequency (ν) of the emitted photon can be calculated using the Rydberg formula for hydrogen-like atoms:

ν = R_∞ × (1/n_f² - 1/n_i²)

Where:

ν = frequency (Hz)
R_∞ = Rydberg constant (1.0973731568508 × 10⁷ Hz)
n_f = final quantum number (3 for n=3)
n_i = initial quantum number (4 for n=4)

For the specific transition from n=4 to n=3, the formula simplifies to:

ν = 1.0973731568508 × 10⁷ × (1/3² - 1/4²) Hz

Example Calculation

Let's calculate the frequency for a transition from n=4 to n=3 in hydrogen:

Identify the quantum numbers: n_i = 4, n_f = 3
Plug values into the formula:

ν = 1.0973731568508 × 10⁷ × (1/9 - 1/16) Hz
Calculate the difference:

1/9 ≈ 0.111111

1/16 = 0.0625

Difference = 0.111111 - 0.0625 = 0.048611
Multiply by the Rydberg constant:

ν ≈ 1.0973731568508 × 10⁷ × 0.048611 ≈ 5.31 × 10⁵ Hz

The frequency of the emitted photon is approximately 531,000 Hz (531 kHz).

Interpreting Results

The calculated frequency corresponds to the energy of the emitted photon. This value is crucial in:

Spectroscopy: Identifying atomic transitions
Astrophysics: Analyzing stellar spectra
Quantum mechanics education: Understanding energy level transitions

Note: For atoms other than hydrogen, the Rydberg constant is adjusted by the atomic number (Z) squared. This calculator uses the hydrogen value (Z=1).

FAQ

What is the difference between frequency and wavelength?

Frequency (ν) is the number of wave cycles per second (Hz), while wavelength (λ) is the distance between wave peaks. They are related by the speed of light (c): c = ν × λ.

Why do electrons only occupy certain energy levels?

Electrons occupy quantized energy levels because quantum mechanics restricts them to specific states that satisfy the Schrödinger equation for the atom's potential.

Can this formula be used for other atoms?

Yes, but the Rydberg constant must be adjusted by the square of the atomic number (Z). For example, for helium (Z=2), use R = R_∞ × Z².