Question

What is the ratio of the energy difference between the ground state and the first excited state for a square infinite potential well of length \(L\) and that for a square infinite potential well of length \(2 L ?\) That is, find \(\left(E_{2}-E_{1}\right)_{L} /\left(E_{2}-E_{1}\right)_{2 L}\)

Short answer

Answer: 3

Step-by-step solution

Energy levels of an infinite potential well

Using the formula for the energy levels of a 1-dimensional infinite potential well: \(E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}\), where n is the energy level, \(\hbar\) is the reduced Planck constant, m is the particle mass, and L is the length of the well.

Energy levels for the well of length L

We will calculate the energy levels for the ground state (n=1) and the first excited state (n=2) for the well of length L: \(E_1 = \frac{1^2 \pi^2 \hbar^2}{2mL^2}\) and \(E_2 = \frac{2^2 \pi^2 \hbar^2}{2mL^2}\)

Energy levels for the well of length 2L

We will calculate the energy levels for the ground state (n=1) and the first excited state (n=2) for the well of length 2L: \(E'_1 = \frac{1^2 \pi^2 \hbar^2}{2m(2L)^2}\) and \(E'_2 = \frac{2^2 \pi^2 \hbar^2}{2m(2L)^2}\)

Energy differences between the ground state and first excited state

We will calculate the energy differences between the ground state and the first excited state for both potential wells: \(\Delta E_L = E_2 - E_1 = \frac{2^2 \pi^2 \hbar^2}{2mL^2} - \frac{1^2 \pi^2 \hbar^2}{2mL^2}\) \(\Delta E_{2L} = E'_2 - E'_1 = \frac{2^2 \pi^2 \hbar^2}{2m(2L)^2} - \frac{1^2 \pi^2 \hbar^2}{2m(2L)^2}\)

Ratio of the energy differences

Now we will find the ratio between the energy differences for the wells of length L and 2L: \(\frac{\Delta E_L}{\Delta E_{2L}} = \frac{\frac{2^2 \pi^2 \hbar^2}{2mL^2} - \frac{1^2 \pi^2 \hbar^2}{2mL^2}}{\frac{2^2 \pi^2 \hbar^2}{2m(2L)^2} - \frac{1^2 \pi^2 \hbar^2}{2m(2L)^2}}\) After simplifying, we get: \(\frac{\Delta E_L}{\Delta E_{2L}} = \frac{3}{1}\) Therefore, the ratio of the energy difference between the ground state and the first excited state for a square infinite potential well of length L and that for a square infinite potential well of length 2L is 3.

Key concepts

Infinite Potential Well and its Characteristics

The concept of an infinite potential well is a fundamental model in quantum mechanics, illustrating how quantization of energy occurs for particles confined within perfectly rigid boundaries.

Imagine a particle that is completely trapped between two walls with an infinite potential energy, implying that it cannot exist outside these walls and has zero potential energy within them. This setup essentially creates a 'box' with hard walls, where the particle is free to move only inside this box. The size of the box is characterized by its length, denoted as 'L'.

The energy states of the particle in this box are not continuous; they are quantized. This means the particle can only possess certain discrete energy values, which are determined using the Schrödinger equation for a given quantum system. The solution to this equation gives rise to a series of allowed energy levels, termed eigenvalues, corresponding to 'standing wave' solutions within the well. These standing waves are the wave functions, or eigenfunctions, of the particle.

Energy Levels of Quantum Systems

The energy levels of quantum systems, such as a particle in an infinite potential well, can be understood through the lens of wave mechanics. In quantum mechanics, these discrete energy levels are a fundamental property and arise due to the boundary conditions that the wave functions must satisfy.

The energy levels are found using the equation \(E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}\), where \(E_n\) represents the energy of the nth state, \(n\) is the principal quantum number, corresponding to the nth energy level, \(\hbar\) is the reduced Planck constant, \(m\) is the mass of the particle, and \(L\) is the length of the potential well.

For example, the first energy level\( (n=1) \) is the ground state, the lowest possible energy state of the system, while \( (n=2) \) represents the first excited state. The energy difference between adjacent levels is not uniform; it increases with an increasing value of \(n\) because it is proportional to \(n^2\). Thus, the structure within a quantum system is such that the energy spacing between levels becomes larger as you go higher in energy.

Quantum State Transitions

Quantum state transitions, also known as quantum jumps, occur when a particle transitions between different energy levels. This can be either by absorption or emission of a quantum of energy, typically in the form of photons.

In the context of the infinite potential well, quantum state transitions are between the quantized energy levels we discussed earlier. When an electron in a quantum well absorbs energy, it may transition from a lower energy level to a higher one, say from the ground state to the first excited state. Conversely, when it loses energy, it 'jumps' back down to a lower energy state, emitting a photon in the process.

The energy of the emitted or absorbed photon corresponds exactly to the energy difference between the two states involved in the transition. This energy difference explains the specific spectral lines observed in quantum systems and provides crucial information about the structure of the system.