# Solving 3D Schrodinger Equation - Explaining 1/X(x) Term

• Master J
In summary, the Schrodinger equation in 3D (time independent) involves a separate Hamiltonian for each Cartesian coordinate, with a term of 1/X(x) etc. that may cause confusion. This occurs because the equations are divided by 1/XYZ to isolate them. A separation in Cartesian coordinates can only work for some potentials, and the derivation starts by assuming an infinite square potential and a separable solution of \Psi (x,y,z) = X(x)Y(y)Z(z).
Master J
The schrodinger equation in 3D (time independent).

Letting Phi = X(x).Y(y).Z(z), and solving as a PDE...

The equation looks pretty much the same except there is a separate Hamiltonian for each of the Cartesian coordinates x y z. However, the 1/X(x) term etc. really confuses me, I don't know where it comes from. Could someone perhaps explain??

ie. H_x = [-(h^2)/2m].[1/X(x)].[(d^2)X(x)/d(X(x))^2] + V(x)
^^^^

where h is representing h-bar, and d the partial derivative.

Cheers guys!

It occurs because you divide through by 1/XYZ to isolate the equations.

But note that using a separation in Cartesian coordinates is not always a viable solution, and will only work for some potentials.

Can you perhaps outline the derivation from the start? It's just clearing it up for me...

It goes something like assume the potential is an infinite square potential

$$V(x,y,z) = \left(\begin{array}{cc}0 if x,y,z < a \\ \infty else$$

We can assume a separable solution $$\Psi (x,y,z) = X(x)Y(y)Z(z)$$

$$\frac{-\hbar^2}{2m} [Y(y)Z(z) \frac{d^2 X}{dx^2}+X(x)Z(z) \frac{d^2 Y}{dy^2}+X(x)Y(y) \frac{d^2 Z}{dz^2}] + V(x,y,z)XYZ = E(XYZ)$$

Then just divide everything by 1/XYZ.

## 1. What is the Schrodinger equation?

The Schrodinger equation is a fundamental equation in quantum mechanics that describes the time evolution of a quantum system. It is used to calculate the probability of finding a particle in a particular location at a specific time.

## 2. What is the 3D Schrodinger equation?

The 3D Schrodinger equation is a version of the Schrodinger equation that takes into account three-dimensional space. It is used to describe the behavior of quantum systems in three-dimensional space, such as atoms and molecules.

## 3. What is the 1/X(x) term in the 3D Schrodinger equation?

The 1/X(x) term in the 3D Schrodinger equation represents the potential energy of the system. It is a function of the position of the particle and influences its behavior and movement.

## 4. How is the 1/X(x) term solved in the 3D Schrodinger equation?

The 1/X(x) term is solved using mathematical techniques such as separation of variables and boundary value problems. These methods involve breaking down the equation into simpler parts and solving them individually.

## 5. Why is solving the 3D Schrodinger equation important?

Solving the 3D Schrodinger equation is important because it allows us to understand the behavior of quantum systems and predict their properties. This has practical applications in fields such as chemistry, materials science, and nanotechnology.

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