Question

What is the ratio of energy difference between the ground state and the first excited state for an infinite square well of length \(L\) to that 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: The ratio of the energy difference between the ground state and the first excited state for an infinite square well of length L to that of length 2L is 1/4.

Step-by-step solution

Find Energy of the Ground State and First Excited State for length L

To find the energy of the ground state (\(E_1\)) for length \(L\), we use the formula above with \(n = 1\). Similarly, to find the energy of the first excited state (\(E_2\)), we use \(n = 2\): $$E_{1}=\dfrac{1^2\hbar^2 \pi^2}{2mL^2}$$ $$E_{2}=\dfrac{2^2\hbar^2 \pi^2}{2mL^2}$$

Calculate the Energy Difference for Length L

Now that we have the energies of both states, the energy difference is given by: $$\Delta E_{L}=E_{2}-E_{1}$$ $$\Delta E_{L}=\dfrac{4\hbar^2 \pi^2}{2mL^2}-\dfrac{1\hbar^2 \pi^2}{2mL^2}$$ $$\Delta E_{L}=\dfrac{3\hbar^2 \pi^2}{2mL^2}$$

Find Energy of the Ground State and First Excited State for length 2L

Similar to Step 1, we find the energies for length \(2L\) as: $$E_{1}=\dfrac{1^2\hbar^2 \pi^2}{2m(2L)^2}$$ $$E_{2}=\dfrac{2^2\hbar^2 \pi^2}{2m(2L)^2}$$

Calculate the Energy Difference for Length 2L

Using the energies from Step 3, we calculate the energy difference: $$\Delta E_{2L}=E_{2}-E_{1}$$ $$\Delta E_{2L}=\dfrac{4\hbar^2 \pi^2}{2m(2L)^2}-\dfrac{1\hbar^2 \pi^2}{2m(2L)^2}$$ $$\Delta E_{2L}=\dfrac{3\hbar^2 \pi^2}{8mL^2}$$

Calculate the Desired Ratio

Finally, we calculate the ratio of the energy differences: $$\dfrac{\Delta E_{L}}{\Delta E_{2L}}=\dfrac{\frac{3\hbar^2 \pi^2}{2mL^2}}{\frac{3\hbar^2 \pi^2}{8mL^2}}$$ Using the fact that $ \dfrac{a}{c} / \dfrac{b}{c}=\dfrac{a}{b}$, $$\dfrac{\Delta E_{L}}{\Delta E_{2L}}=\dfrac{2mL^2}{8mL^2}=\dfrac{1}{4}$$ Therefore, the ratio of the energy difference between the ground state and the first excited state for an infinite square well of length \(L\) to that of length \(2L\) is \(\dfrac{1}{4}\).

Key concepts

Infinite Square Well

In quantum mechanics, the infinite square well is a model used to describe how particles, like electrons, behave in a confined space. Imagine a particle that is completely trapped in a box with walls so high that it cannot escape. This box is the infinite square well. The particle can move freely inside the box but cannot pass through the walls. This simple yet powerful concept helps us understand the quantum behavior of particles.

The infinite square well is often represented as a one-dimensional box with length \( L \). Because of the well's infinite depth, it imposes boundary conditions: the particle's wave function must go to zero at the walls.

• This boundary condition results in only specific, quantized energies or states being allowed for the particle.
• These energies depend on the width of the well \( L \) and the mass of the particle \( m \).

The energy levels in the well are described by a simple formula, where energy increases with the quantum number \( n \). By understanding these principles, students can explore more complex quantum systems.

Energy States

Energy states in quantum mechanics refer to the specific levels of energy that a quantum system can have. In the case of the infinite square well, these states are discrete, meaning they can only take on certain values. The ground state (\( E_1 \)) is the lowest energy state, and it occurs when the particle is in the least energetic arrangement possible.

As the particle moves to higher energy states, like the first excited state (\( E_2 \)), it occupies levels with increasing energy. The formula for the energy of these states is given by:

• \( E_n = \frac{n^2 \hbar^2 \pi^2}{2mL^2} \)

Here, \( n \) is the quantum number, \( \hbar \) is the reduced Planck's constant, and \( m \) is the mass of the particle. What's interesting is:

• As \( n \) increases, the particle's energy increases dramatically.
• The energy levels are equidistant in the infinite square well.

This quantization is central to understanding quantum systems, as it contrasts with classical systems where energy is continuous.

Quantum Energy Levels

Quantum energy levels are the distinct energy values that a quantum system, like an electron in an infinite square well, can have. These levels are determined by the wave-like properties of particles described by quantum mechanics. In an infinite square well, energy levels are not influenced by external conditions but are entirely set by the well's size and the particle's mass.

Let's take a specific case to learn more. For a box of length \( L \), the energy difference between two consecutive energy levels, such as between the ground state and the first excited state, can be calculated using their individual energy formulas. The difference for length \( L \) is given by:

• \( \Delta E_L = \frac{3\hbar^2 \pi^2}{2mL^2} \)

While for a box of double the length \( 2L \), it is:

• \( \Delta E_{2L} = \frac{3\hbar^2 \pi^2}{8mL^2} \)

The ratio of these energy differences gives insight into how energy spacing changes with the size of the well. The ratio \( \frac{1}{4} \) suggests that doubling the well's length decreases the energy difference between states quite significantly. Understanding these concepts is crucial for grasping the vast and fascinating world of quantum mechanics.