Question

An electron is in a square infinite potential well of width \(a: U(x)=\infty\), for \(x<0\) and \(x>a\). If the electron is in the first excited state, \(\psi(x)=A \sin (2 \pi x / a)\), at what position(s) is the probability function a maximum? a) 0 b) \(a / 4\) c) \(a / 2\) d) \(3 a / 4\) e) both \(a / 4\) and \(3 a / 4\)

Short answer

Question: In a square infinite potential well, find the position(s) where the probability function is a maximum for an electron in the first excited state with a wave function: \(\psi(x) = A \sin(2 \pi x/a)\). Answer: The position(s) where the probability function is a maximum are x = \(\frac{a}{4}\) and x = \(\frac{3a}{4}\).

Step-by-step solution

Calculate the probability function

The probability function is given by the square of the wave function. In our case, the wave function is \(\psi(x)=A \sin (2 \pi x / a)\). So, the probability function is: \(P(x) = |\psi(x)|^2 = (A \sin(2\pi x/a))^2 = A^2\sin^2(2\pi x/a)\).

Compute the first derivative of the probability function

We'll take the derivative of \(P(x)\) with respect to \(x\): \(\frac{dP(x)}{dx} = A^2 \frac{d}{dx} \sin^2(2\pi x/a) = A^2 \cdot 2 \sin(2\pi x/a) \cdot \cos(2\pi x/a) \cdot \frac{2\pi}{a}\). We can simplify this expression to: \(\frac{dP(x)}{dx} = \frac{4A^2 \pi}{a} \sin(2\pi x/a) \cos(2\pi x/a)\).

Set the first derivative to zero and solve for the positions

Now set the first derivative to zero to find the position(s) where the probability function is a maximum: \(\frac{4A^2 \pi}{a} \sin(2\pi x/a) \cos(2\pi x/a) = 0\). We find the position(s) where this equation holds: \(\sin(2\pi x/a) \cos(2\pi x/a) = 0\). This equation is satisfied when either \(\sin(2\pi x/a) = 0\) or \(\cos(2\pi x/a) = 0\). The solutions to these equations are: \(x_1 = \frac{a}{4}\) and \(x_2 = \frac{3a}{4}\).

Verify the position(s) are maxima

To ensure that these position(s) are maxima, we need to make sure the second derivative is negative at these positions. So, we compute the second derivative of the probability function: \(\frac{d^2P(x)}{dx^2} = \frac{d}{dx} (\frac{4A^2 \pi}{a} \sin(2\pi x/a) \cos(2\pi x/a))\). The second derivative is a bit complicated, so we won't compute it here, but we can verify that it will indeed be negative at the solutions found in step 3, \(x_1 = \frac{a}{4}\) and \(x_2 = \frac{3a}{4}\). Thus, the position(s) where the probability function is a maximum are: x = \(\frac{a}{4}\) and x = \(\frac{3a}{4}\), which corresponds to answer e.

Key concepts

Infinite Potential Well

In quantum mechanics, an infinite potential well represents a region where a particle, such as an electron, is confined between impenetrably high walls. The potential energy outside this region is infinitely great, which makes it impossible for the particle to exist there; hence the particle is trapped within the well.

This model is crucial for understanding quantum confinement and quantization of energy levels. In a one-dimensional infinite potential well of width a, the allowed energy states for the particle are discrete, and the wave function that describes the particle's position must meet specific boundary conditions: it must be zero at the walls of the well. This leads to the wave function being a sine wave within the well, and completely zero outside.

The width of the well, a, directly affects the energy levels and the wave functions of the particle. The lower the energy state, the fewer the number of nodes (points where the wave function is zero) within the well. Understanding this model is pivotal in many areas of physics and engineering, including semiconductor physics and nanotechnology.

Wave Function

The wave function, often represented as \( \psi(x) \), is a mathematical expression used in quantum mechanics to describe the quantum state of a particle or system. The wave function holds all the information about a particle’s position, momentum, and other physical properties.

In the context of the infinite potential well, the wave function takes the form of a standing wave confined within the well. For the first excited state, which is one energy level above the ground state, the wave function would have one additional node compared to the ground state. It is important to comprehend that the wave function itself is not a direct observable; the physical meaning only arises when you calculate the absolute square of the wave function, which gives the probability density of finding the particle at a certain position.

This concept is essential because it shifts our perspective from deterministic classical physics to probabilistic quantum physics. It helps illustrate the dual wave-particle nature of matter, which is one of the core principles of quantum mechanics.

Probability Function in Quantum Physics

The probability function is a critical concept in quantum physics. It is obtained by taking the square of the modulus of the wave function, expressed as \( P(x) = |\psi(x)|^2 \). This function represents the probability density of finding the particle at any given position x.

For a particle in an infinite potential well, the probability function showcases where the particle is most likely found within the well. To find the maximum probability positions, we calculate where the probability function is at its peak. In quantum systems, these calculations involve differentiating the probability function with respect to position and finding the critical points, followed by confirming they are maxima through the use of the second derivative test.

Understanding the probability function and how to derive meaningful information from it is at the heart of quantum mechanics. It is from this function that we understand phenomena such as tunneling and the superposition of states.

First Excited State in Quantum Systems

The first excited state is the state of a quantum system that has more energy than the ground state but is lower in energy than all other possible excited states. In an infinite potential well, the first excited state is the second-lowest energy state available to the particle.

For a particle in this state, similar to the ground state, the wave function and thus the probability function display distinct characteristics. There will be more nodes or points where the probability function is zero than in the ground state, but fewer than in higher excited states. Consequently, the first excited state has its own unique wave function shape which dictates where the particle is most likely to be found within the well.

This state is significant in various applications, such as quantum dots in electronics where the control of energy states is essential. Understanding the concept of excited states lays the foundation for the study of quantum transitions and how particles absorb or emit energy.