Homework Help is Here – Start Your Trial Now!
Science
Advanced Physics
Start with the differential equation for ψ within the well for E = 0, as provided in the second image. What is the most general solution to this second-order differential equation? Show that the reqiurement that the wave function vanishes at the boundaries of the well leads to ψ = 0.

Start with the differential equation for ψ within the well for E = 0, as provided in the second image. What is the most general solution to this second-order differential equation? Show that the reqiurement that the wave function vanishes at the boundaries of the well leads to ψ = 0.

icon
Related questions
Question

Show that there is no solution to the time-independent Schrodinger equation for a particle in the infinte square well for E = 0, as provided in the first image.

Follow these steps to prove that there is no solution:

  • Start with the differential equation for ψ within the well for E = 0, as provided in the second image.
  • What is the most general solution to this second-order differential equation?
  • Show that the reqiurement that the wave function vanishes at the boundaries of the well leads to ψ = 0.
The image presents a mathematical expression for a potential function, \( V(x) \), defined as follows: \[ V(x) = \begin{cases} 0 & \text{for } 0 < x < L \\ \infty & \text{elsewhere} \end{cases} \] This describes a potential well with boundaries at \( x = 0 \) and \( x = L \). The potential is zero within this interval, indicating a region where a particle could move freely, and infinite outside of it, representing impenetrable barriers.
Transcribed Image Text:The image presents a mathematical expression for a potential function, \( V(x) \), defined as follows: \[ V(x) = \begin{cases} 0 & \text{for } 0 < x < L \\ \infty & \text{elsewhere} \end{cases} \] This describes a potential well with boundaries at \( x = 0 \) and \( x = L \). The potential is zero within this interval, indicating a region where a particle could move freely, and infinite outside of it, representing impenetrable barriers.
The image shows the following mathematical equation: \[ -\frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} = 0 \] This equation is a simplified version of the Schrödinger equation, often encountered in the context of quantum mechanics. It describes the behavior of a free particle (i.e., a particle not subjected to any forces) at the quantum level. **Explanation of the symbols:** - \(\hbar\) is the reduced Planck's constant, a fundamental physical constant used in quantum mechanics. - \(m\) represents the mass of the particle. - \(\psi\) is the wave function of the particle, which contains all the information about the system. - \(\frac{d^2 \psi}{dx^2}\) denotes the second derivative of the wave function \(\psi\) with respect to position \(x\). This particular form implies that the second derivative of the wave function with respect to position is zero, which usually suggests a solution involving a linear function of position, indicating no change in the potential energy of the system.
Transcribed Image Text:The image shows the following mathematical equation: \[ -\frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} = 0 \] This equation is a simplified version of the Schrödinger equation, often encountered in the context of quantum mechanics. It describes the behavior of a free particle (i.e., a particle not subjected to any forces) at the quantum level. **Explanation of the symbols:** - \(\hbar\) is the reduced Planck's constant, a fundamental physical constant used in quantum mechanics. - \(m\) represents the mass of the particle. - \(\psi\) is the wave function of the particle, which contains all the information about the system. - \(\frac{d^2 \psi}{dx^2}\) denotes the second derivative of the wave function \(\psi\) with respect to position \(x\). This particular form implies that the second derivative of the wave function with respect to position is zero, which usually suggests a solution involving a linear function of position, indicating no change in the potential energy of the system.
Expert Solution
steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer