# Exploring Shankar's PQM: Reducing to Schroedinger's Eqn

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• hideelo
In summary, Shankar discusses the path integral formulation and its reduction to Schroedinger's equation on page 230 of his PQM (1994 edition). He argues that the first exponential in the equation will oscillate excessively due to the small value of ##\epsilon \hbar##. To control this, he suggests restricting the range of the variable of integration, ##x'##, to satisfy ##\frac{m \eta^2}{2\epsilon \hbar} \leq \pi##. However, he then changes the variable of integration to ##\eta## and integrates over the entire range, which seems contradictory. But this is justified by the small value of ##\epsilon \hbar##, as the
hideelo
On page 230 in Shankar's PQM (1994 edition) he is trying to show that the path integral formulation reduces to Schroedinger's eqn. The equation he comes up against is the following

$$\psi (x,\epsilon) = \sqrt{\frac{m}{2 \pi i \epsilon \hbar}}\int_{-\infty}^{\infty}dx' \psi (x',0) \exp\left[\frac{im(x-x')^2}{2\epsilon \hbar} \right] \exp\left[\frac{-i\epsilon}{ \hbar} V\left( \frac{x+x'}{2},0\right) \right]$$

He makes the argument that the first exponential is going to oscillate like hell because ##\epsilon \hbar## is so small. He says that in order to keep this under control we need to restrict the range of x' so that

$$\frac{m \eta^2}{2\epsilon \hbar} \leq \pi$$
where
$$\eta = (x-x')$$

which I follow. He then changes the variable of integration from x' to ##\eta##, no big deal. He expands everything inside the integral to second order in ##\eta## because that corresponds to first order in ##\epsilon##. I'm still on board. He then integrates over ##\eta## from ##-\infty## to ##\infty## and this is where he loses me. What happened to ##\frac{m \eta^2}{2\epsilon \hbar} \leq \pi## ? The way he expressed everything inside the integral assumed that ##\eta## is small. So why is he integrating over the whole range of ##\eta##?

It doesn't matter, because ##\epsilon \hbar## is so small. That's the usual way to evaluate an integral approximately with the method of steepest descent. You usually get an asymptotic series with that technique.

## 1. What is Shankar's PQM?

Shankar's PQM (Principles of Quantum Mechanics) is a textbook written by Ramamurti Shankar that provides an in-depth exploration of quantum mechanics, which is the branch of physics that deals with the behavior of particles at a microscopic level.

## 2. What does it mean to reduce to Schroedinger's equation?

Reducing to Schroedinger's equation means simplifying a complex quantum mechanical system into a single equation, known as the Schroedinger equation. This equation describes the behavior of wave functions, which represent the probability of finding a particle at a certain location.

## 3. Why is reducing to Schroedinger's equation important?

Reducing to Schroedinger's equation allows us to mathematically describe and predict the behavior of particles at a microscopic level. This is essential for understanding and developing technologies such as quantum computing and nanotechnology.

## 4. How does Shankar's PQM explore the concept of reducing to Schroedinger's equation?

Shankar's PQM provides a comprehensive and detailed explanation of the principles behind reducing to Schroedinger's equation. It covers topics such as wave mechanics, operators, and the mathematical foundations of quantum mechanics, all of which are necessary for understanding and applying the concept of reducing to Schroedinger's equation.

## 5. Is Shankar's PQM suitable for beginners in quantum mechanics?

No, Shankar's PQM is a more advanced textbook and is better suited for those with a solid foundation in classical mechanics and mathematics. It is commonly used as a graduate-level textbook in physics courses.

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