Question

An electron is in an infinite square 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 is the probability function a maximum? a) 0 b) \(a / 4\) c) \(a / 2\) d) \(3 a / 4\) and \(3 a / 4\) e) at both \(a / 4\)

Short answer

Answer: c) \(a / 2\)

Step-by-step solution

Calculate the square of the wave function

First, we find the probability function by squaring the given wave function. The probability function \(P(x)\) can be written as: \(P(x) = |\Psi(x)|^2 = (A \sin(2\pi x / a))^2 = A^2\sin^2(2\pi x / a)\)

Differentiate the probability function with respect to x

Now, we calculate the derivative of the probability function with respect to x: \(\frac{dP(x)}{dx} = \frac{d}{dx}(A^2\sin^2(2\pi x / a))\) Using the chain rule for differentiation, we can break down this derivative as follows: \(\frac{dP(x)}{dx} = 2A^2 \sin(2\pi x / a) \cdot \cos(2\pi x / a) \cdot \frac{2\pi}{a}\)

Set the derivative equal to zero and solve for x

To find the maximum of the probability function, we set the derivative equal to zero: \(0 = 2A^2 \sin(2\pi x / a) \cdot \cos(2\pi x / a) \cdot \frac{2\pi}{a}\) First, we note that \(A^2\) and \(\frac{2\pi}{a}\) are constants, which do not affect the condition for the function maximum. So we can remove them from the equation: \(0 = \sin(2\pi x / a) \cdot \cos(2\pi x / a)\) Now, we need to find the values of x that make the equation true. One way to do this is by using the double-angle trigonometric identity for sine: \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\) Comparing this to our equation, we see that: \(\theta = \pi x / a\) Thus, we can rewrite our equation as: \(0 = 2\sin(\pi x / a) \cos(\pi x / a)\) For this equation to be true, either \(\sin(\pi x / a)\) or \(\cos(\pi x / a)\) must be equal to zero. Let's solve for x in both cases: Case 1: \(\sin(\pi x / a) = 0\) \(\pi x / a = n\pi\) \(x = na\), where n is an integer. Case 2: \(\cos(\pi x / a) = 0\) \(\pi x / a = (2n+1)\pi/2\) \(x = (2n+1)a/2\), where n is an integer. Now, we need to choose the x values from these two cases that are within the range of the infinite square well (0 < x < a). According to case 1, x can be equal to 0 or a, but these are not within the limit of the well. In case 2, for n=0, we have x = a/2 which is within the limit. So, we choose the x value accordingly.

Determine the answer

Having analyzed both cases, we find that the probability function, \(P(x)\), is at a maximum when \(x = a / 2\). The correct answer is: c) \(a / 2\)

Key concepts

Infinite Square Well

In quantum mechanics, the infinite square well is a fundamental model used to understand particle behavior in confined spaces. It features an impenetrable barrier with infinite potential energy outside a defined region. Inside this region, potential energy is set to zero, allowing particles such as electrons to move freely. The walls of the well are assumed to be infinitely high, making it impossible for particles to escape.

• The finite width of the well, typically termed as 'a', establishes the region in which the particle is confined.
• Outside this range, the wave function, which describes the particle's state, drops to zero.
• Solutions to particle behavior in this potential include quantized energy levels, which means a particle can only possess certain energy values.

The infinite square well makes it easier to predict behaviors like energy levels and states, key aspects of quantum mechanics.

Wave Function

The wave function, represented by \( \Psi(x) \), is a mathematical description in quantum mechanics that summarizes the quantum state of a system. Specifically, it provides information about the probability amplitude of finding a particle at position x.

• The wave function is a core concept in quantum mechanics, serving as a foundation for understanding properties and behaviors of particles.
• For an infinite square well, solutions to the wave function take a sinusoidal form like \( A \sin(n \pi x / a) \).
• The amplitude A is a normalization constant ensuring that the total probability of finding the particle within the well is 1.

Knowing the wave function is key to calculating other crucial quantities, such as the probability function.

Probability Function

The probability function determines where a particle is most likely to be found in space. In quantum mechanics, it's derived from the wave function by taking its absolute square, expressed as \( P(x) = |\Psi(x)|^2 \).

• This quantifies the likelihood of finding a particle at various positions within the well.
• The probability function needs to be normalized, meaning the integral of \( P(x) \) over all space equals one, ensuring all possible positions are considered.
• Maxima of this function indicate points where the particle is most likely to be found, making them crucial for studying particle dynamics.

In the exercise, determining where \( P(x) \) is maximized provides insight into the electron's most probable location within the potential well.

First Excited State

The first excited state of a particle in a quantum well represents the state with the second-lowest energy level above the ground state. It is described by a distinct wave function with its own properties and behavior.

• For an infinite square well, this state corresponds to the solution \( \Psi(x) = A \sin(2 \pi x / a) \).
• This function shows a higher frequency with additional nodes compared to the ground state, which generally means more wave peaks and troughs inside the well.
• The first excited state is significant due to its energy and probability distribution differences compared to the ground state, impacting how particles behave under excitation.

Understanding excited states is key for applications involving energy transitions and quantum behaviors.