Askiitians Tutor Team
Last Activity: 16 Days ago
In quantum mechanics, when we introduce a small perturbation to a system, we can use perturbation theory to find how this affects the energy levels of the system. In the case of a one-dimensional infinite potential well, the ground state energy can be modified by a small dip in potential energy. Let’s break down the steps to derive the first-order correction to the ground state energy and evaluate it for a specific case.
Understanding the Infinite Potential Well
In an infinite potential well, the potential energy is defined as:
The wave functions for the ground state (n=1) in this well are given by:
The corresponding energy for the ground state is:
Introducing the Perturbation
Now, let’s introduce a small dip in the potential energy, , which we can represent as:
for (where )
Outside this region, the potential remains zero. The first-order correction to the energy can be calculated using the formula:
For the ground state (n=1), this becomes:
Calculating the First-Order Energy Correction
Substituting the wave function and the potential into the integral, we have:
Since , we can write:
Now, simplifying the integral:
Evaluating the Integral
To evaluate the integral, we can use the identity:
Thus, we have:
Substituting this back into our expression for , we can find the first-order correction to the ground state energy.
Final Expression
After performing the calculations, we can summarize the first-order energy correction as:
Now, substituting and evaluating the integral will give you the specific numerical value for the energy correction based on the chosen .
This approach illustrates how perturbation theory allows us to analyze small changes in a quantum system and understand their impact on energy levels. If you have any specific values for or further questions about the calculations, feel free to ask!