The method of steepest descent is a mathematical technique used to approximate the solution to an integral or sum by finding the stationary point of an exponential function. In the context of a quartic oscillator, this method can be used to calculate the ground state energy of the system.
The method of steepest descent involves finding the saddle point of the exponential function. In the case of a quartic oscillator, this involves solving the Euler-Lagrange equation to find the critical points of the action functional. The saddle point corresponds to the ground state energy of the system.
This method provides a simple and efficient way to approximate the ground state energy of a quartic oscillator without having to solve the Schrödinger equation. It also allows for the calculation of higher-order corrections to the ground state energy, providing a more accurate result.
The method of steepest descent may not give an accurate result if the saddle point of the exponential function is not close to the actual minimum. This can happen in cases where there are multiple saddle points or when the potential is highly asymmetric.
The method of steepest descent is closely related to other approximation methods, such as the WKB approximation and the variational method. It can be seen as a generalization of the WKB method, and it provides a variational estimate for the ground state energy. It also provides a basis for perturbation methods, which use the saddle point as a starting point for calculating higher-order corrections.