Question

The wavelength of an electron in an infinite potential is $\alpha /2,$ where $\alpha$ is the width of the infinite potential well. Which state is the electron in? a) $n=3$ b) $n=6$ c) $n=4$ d) $n=2$

Short answer

Answer: c) $n=4$

Step-by-step solution

Write down the formula for the wavelength of an electron in an infinite potential well.

The wavelength of an electron in an infinite potential well can be expressed as ${\lambda }_n=\frac{2\alpha }{n}$, where ${\lambda }_n$ denotes the wavelength of an electron in the nth state, $\alpha$ is the width of the potential well, and n is an integer representing the quantum state.

Plug in the given wavelength.

We are given that $\lambda =\frac{\alpha }{2}$. Now we want to find the value of n for which this equation is true.

Solve for n.

We want to find n, such that $\frac{\alpha }{2}=\frac{2\alpha }{n}$. We can rearrange this equation as follows: $$n=\frac{2\alpha }{\frac{\alpha }{2}}$$

Simplify the equation.

Now, we have: $$n=\frac{2\alpha \times 2}{\alpha }$$

Cancel out the common factors.

In the expression above, we can cancel out the alpha terms: $$n=\frac{2\cancel{\alpha }\times 2}{\cancel{\alpha }}$$

Find the value of n.

Thus, we are left with: $$n=2\times 2=4$$

Match the answer with the given options.

We found that the electron is in state n = 4. So, the correct answer is: c) $n=4$

Key concepts

Infinite Potential Well

An infinite potential well is a fundamental concept in quantum mechanics. Imagine a narrow region where an electron is trapped and cannot escape due to extremely high potential energy barriers on either side. This leads to the electron being confined in a "well" with infinite walls.

This model is significant because it illustrates how electrons behave in confined spaces, like around an atom or in certain semiconductor materials. When an electron is in an infinite potential well, it cannot have just any energy; it must have specific, discrete energy levels corresponding to distinct quantum states. These energy levels mean that the electron can "jump" between levels, but it requires energy input or output to do so.

• An infinite potential well is a hypothetical enclosure with infinitely high walls, which theoretically prevents the electron from leaving the well.
• This concept helps to understand boundary conditions and wave functions concerning electron behavior.

Understanding an infinite potential well is crucial for studying and predicting quantum behavior in other more complex systems.

Electron Wavelength

Electron wavelength is a vital concept when dealing with quantum mechanics. Picture it as the "length" of a wave associated with an electron's motion. Due to the wave-particle duality, we can't think of electrons just as tiny particles. Instead, they also have wave-like characteristics.

In the context of an infinite potential well, each quantum state an electron can occupy corresponds to a specific wavelength. The wavelength is inversely proportional to the electron's energy level. So, the higher the energy state, the shorter the wavelength. It's like plucking a guitar string at different tensions—each pluck (or state) produces a different note (or wavelength).

• The electron's wavelength ( ext{λ} ) in a potential well is given by ext{λ} = rac{2 ext{α}}{n} , where ext{α} is the width of the well, and n is the quantum state.
• Shorter wavelengths correspond to higher frequencies and higher energy states.

This understanding of electron wavelength is fundamental in comprehending how quantum states and energy levels work.

Quantum States

Quantum states refer to the specific, allowable energy levels that an electron can have within a potential well. Instead of moving smoothly between energy levels like classical objects, electrons jump between discrete states in quantum mechanics.

Each quantum state in an infinite potential well is denoted by the integer ext{n} , known as the principal quantum number. The energy and the corresponding wavelength depend on these quantum states. So, for an electron trapped in such a well, only specific quantum states are permissible: ext{Λ}_n = rac{2 ext{α}}{n} is associated with these states.

• Each quantum state has a particular energy level, calculated using n.
• n=1 corresponds to the lowest energy state or ground state, while higher values of n correspond to excited states.
• The electron must absorb or release energy to transition between quantum states.

The notion of quantum states helps explain phenomena like electron transitions and spectral lines, crucial for technologies like lasers and understanding atomic structures.

Wave-Particle Duality

Wave-particle duality is a core principle of quantum mechanics that describes how electrons and other particles exhibit both wave-like and particle-like properties, depending on the situation.

When electrons are confined within an infinite potential well, their wave-like nature becomes evident. They form standing waves within the well, where certain "wavelengths" or states are allowed due to the boundary conditions. This duality is why electrons can occupy discrete quantum states while also displaying interference patterns like waves in experiments such as the double-slit experiment.

• Wave-particle duality underpins quantum behavior, illustrating the unique nature of electrons as they don't follow classical physics.
• This concept helps explain why even seemingly solid particles like electrons have measurable wavelengths.

This principle is essential in understanding the basics of quantum mechanics, demonstrating the complexity and peculiar nature of particles at the quantum level.