# How Is the Schrödinger Equation Derived from the Hamilton-Jacobi Equation?

• hunc
In summary: But we know that \frac{(\nabla S)^2}{2m} + V = -\frac{d}{dt} S, so these two terms are equal. Therefore, the leading order terms of \frac{-\hbar^2}{2m} \nabla^2 \phi and i \hbar \frac{d}{dt} \phi are equal, and we get the Schrödinger equation.The wiki page you mentioned is talking about a different equation, the nonlinear Schrödinger equation, which is not the same as the linear Schrödinger equation. In summary, the book is trying to derive the linear Schrödinger equation by finding an evolution
hunc

## Homework Statement

I am reading Mathematical Concepts of Quantum Mechanics (Stephen J. Gustafson, Israel Michael Sigal. Second edition). The book would like to find an evolution equation which would lead to the Hamilton-Jacobi equation
$$\frac{\partial S}{\partial t}=-h(x, \nabla S)$$
in the way the wave equation led to the eikonal one. The book also says that ##\phi (x, t) = a(x, t) \exp( i S(x,t)/\hbar)##. So I express ##S(x,t)## using ##\phi (x,t)## and substitute back to the Hamilton-Jacobi equation, taking ##h (x, \nabla S) = \frac{1}{2m}|\nabla S|^2+V(x)##.

The book means to take the leading terms when ##\hbar## small compared to a typical classical action ##S## and restore Schrodinger equation. I am kind of lost during the derivation.

## Homework Equations

After the substitution, I have
$$i\hbar\partial_t \phi=-\frac{\hbar^2}{2m}[(\frac{\nabla \phi}{\phi}-\frac{\nabla a}{a})^2-\frac{2im\partial_t a}{a\hbar}]\phi+V(x)\phi.$$
Comparing with Schrodinger equation, I figure that the leading term of
$$[(\frac{\nabla \phi}{\phi}-\frac{\nabla a}{a})^2-\frac{2im\partial_t a}{a\hbar}]\phi$$
should equal to ##\Delta_x \phi##, but don't know how.

## The Attempt at a Solution

I am not sure what to search for the problem, but wiki have something on this. A nonlinear variant of the Schrödinger equation is expressed as
$$i\hbar\partial_t \phi=-\frac{\hbar^2}{2m}\frac{(\nabla \phi)^2}{\phi}+V(x)\phi.$$
I am not sure what a nonlinear Schrödinger equation is after realizing it's not the same thing as the Schrödinger equation.

The book's goal seems to be the linear Schrödinger equation. Even though I do see how to obtain the nonlinear Schrödinger equation, I am not sure why ##(\frac{\nabla \phi}{\phi})^2## is a leading term. Could someone help me with this?

Thanks!

You can derive it in the following way:

1. Let $\phi = a e^{\frac{iS}{\hbar}}$
2. Compute $\frac{-\hbar^2}{2m} \nabla^2 \phi$ in terms of $a$ and $S$ and only keep the lowest-order terms, in powers of $\hbar$
3. Similarly, compute $i \hbar \frac{d}{dt} \phi$ in terms of $a$ and $S$ and keep the lowest-order terms in powers of $\hbar$.
4. Now, use the fact that $\frac{(\nabla S)^2}{2m} + V = -\frac{d}{dt} S$ to show that the results of 2 and 3 are equal (to lowest order in $\hbar$)

stevendaryl said:
You can derive it in the following way:

1. Let $\phi = a e^{\frac{iS}{\hbar}}$
2. Compute $\frac{-\hbar^2}{2m} \nabla^2 \phi$ in terms of $a$ and $S$ and only keep the lowest-order terms, in powers of $\hbar$
3. Similarly, compute $i \hbar \frac{d}{dt} \phi$ in terms of $a$ and $S$ and keep the lowest-order terms in powers of $\hbar$.
4. Now, use the fact that $\frac{(\nabla S)^2}{2m} + V = -\frac{d}{dt} S$ to show that the results of 2 and 3 are equal (to lowest order in $\hbar$)
Thanks! It was my first attempt, which never really got carried out. I thought ##\nabla^2\phi## can bring in ##1/\hbar^2## and ##\partial_t \phi## only ##1/\hbar##... And I just go through it and all is fine.
And now I kind of want to ask what's the story about the wiki and the equation
$$i\hbar\partial_t \phi=-\frac{\hbar^2}{2m}\frac{(\nabla \phi)^2}{\phi}+V(x)\phi.$$

Well, if you let $\phi = a e^{\frac{i}{\hbar} S}$, then

$\phi^* (-\frac{\hbar^2}{2m} \nabla^2 \phi) = -\frac{\hbar^2}{2m} a \nabla^2 a + \frac{\hbar^2}{2m} (\nabla a)^2 - \frac{i \hbar}{2m} (\nabla^2 S) a^2 + \frac{1}{2m} (\nabla S)^2 a^2$

$\frac{\hbar^2}{2m} |\nabla \phi|^2 = \frac{\hbar^2}{2m} (\nabla a)^2 + \frac{1}{2m} (\nabla S)^2 a^2$

So the difference between them is $-\frac{\hbar^2}{2m} a \nabla^2 a - \frac{i \hbar}{2m} (\nabla^2 S) a^2$.

The Schrodinger equation is a fundamental equation in quantum mechanics that describes the evolution of a quantum state over time. It was first derived by Erwin Schrodinger in 1925 and is based on the principles of wave mechanics.

To derive the Schrodinger equation, we start with the Hamilton-Jacobi equation, which is a classical equation that describes the evolution of a classical system. In the Hamilton-Jacobi equation, the quantity S represents the classical action, which is a measure of the energy of the system. The book suggests that we can use the concept of wave mechanics to express S in terms of a wave function, ψ.

Substituting this expression for S into the Hamilton-Jacobi equation, we get an equation that involves the wave function ψ and its derivatives. This equation is known as the Schrodinger equation, and it describes the evolution of the quantum state over time.

The leading term in the Schrodinger equation is the Laplacian operator, which is represented as ∇^2 or Δ. This term represents the kinetic energy of the system and is related to the momentum of the particles in the system. The book suggests that this term can be obtained from the expression for S by taking the leading terms when ℏ is small compared to a typical classical action S.

In summary, the Schrodinger equation is derived by substituting the expression for S in terms of the wave function ψ into the classical Hamilton-Jacobi equation. The leading term in the resulting equation is the Laplacian operator, which represents the kinetic energy of the system. This equation is known as the Schrodinger equation and it describes the evolution of the quantum state over time.

## 1. What is the Schrodinger equation?

The Schrodinger equation is a mathematical equation that describes how the wave function of a quantum system evolves over time. It is a key equation in quantum mechanics and is used to calculate the probability of finding a particle in a certain position or state.

## 2. Who created the Schrodinger equation?

The Schrodinger equation was developed by Austrian physicist Erwin Schrodinger in 1926. He was trying to find a way to describe the behavior of electrons in atoms, and his equation became a fundamental tool in quantum mechanics.

## 3. How is the Schrodinger equation derived?

The Schrodinger equation is derived using mathematical principles and concepts from classical mechanics and electromagnetism. It is based on the principle of superposition, which states that a quantum system can exist in multiple states at the same time.

## 4. What is the significance of the Schrodinger equation?

The Schrodinger equation is a fundamental equation in quantum mechanics and is used to make predictions about the behavior of particles at the microscopic level. It has been successfully applied to a wide range of physical systems, from atoms and molecules to subatomic particles and large-scale systems like superconductors.

## 5. How is the Schrodinger equation used in practical applications?

The Schrodinger equation is used in many practical applications, such as in the development of new materials and technologies, designing drugs and medicines, and understanding chemical reactions. It is also a key tool in fields like quantum computing and quantum cryptography.

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