A one-dimensional harmonic oscillator is perturbed by an extra potential energy \(b x^{3} .\) Calculate the change in each energy level to second order in the perturbation.

To understand the effect of a small additional potential on a quantum system, we use perturbation theory. This theory helps us calculate changes in the energy levels without solving the system from scratch. The primary idea is to start with a known system, such as the harmonic oscillator, and apply a small, manageable change. By doing so, we can determine corrections to the energy levels due to this perturbation. Perturbation theory breaks down the correction process into orders: first order and second order corrections. Each order gives us a more accurate approximation of the new energy levels.

In the case of our one-dimensional harmonic oscillator, adding the potential energy term, \(V_1(x) = b x^3\), introduces a perturbation. The energy levels of the unperturbed harmonic oscillator are given by \(E_n^0 = \left(n + \frac{1}{2}\right)\hbar \omega\), where \(n\) is a non-negative integer and \(\hbar\) is the reduced Planck's constant.
First order corrections are calculated using the formula \(E_n^1 = \langle n | V_1 | n \rangle\). However, in this specific scenario, the first order correction is zero because \(\langle n | x^3 | n \rangle = 0\) due to the symmetry properties of the wave functions.
The second order corrections are more involved and require the calculation of off-diagonal matrix elements and their sums.

Matrix elements are essential in quantum mechanics, especially for perturbation theory. They help us calculate the transition probabilities between states. For the given perturbation \(V_1(x) = b x^3\), the matrix elements \(\langle m | x^3 | n \rangle\) are needed. These elements are often easier to compute using ladder operators. These ladder operators, also known as creation and annihilation operators, simplify the calculations by allowing us to express complex functions in a more manageable form.
For the harmonic oscillator, ladder operators are used to determine the matrix elements efficiently, as they naturally account for the symmetries and properties of the harmonic oscillator wave functions.

Ladder operators, \(a\) and \(a^\dagger\), are tools used to describe the quantum states of the harmonic oscillator. The annihilation operator, \(a\), lowers the energy state of the system, while the creation operator, \(a^\dagger\), raises it. These operators help find matrix elements of position and momentum in a harmonic oscillator. For instance, the position operator \(x\) can be expressed in terms of these operators: \(x = \sqrt{\frac{\hbar}{2m\omega}} (a + a^\dagger)\). By expressing \(x^3\) in terms of \(a\) and \(a^\dagger\), we can systematically calculate the required matrix elements. This helps in computing the second order energy corrections more accurately.

Quantum mechanics is the study of particles at the microscopic level, where classical physics no longer applies. In this theory, energy levels are quantized, meaning particles can only exist in specific states with distinct energy levels. The harmonic oscillator is a fundamental model in quantum mechanics, describing particles in a quadratic potential well. When perturbations such as additional potential energy terms are introduced, quantum mechanics provides the tools to understand how these perturbations alter the system's properties. Using perturbation theory and tools like matrix elements and ladder operators, we can delve deeper into these changes and predict the behavior of quantum systems more accurately.