Seperation of variables, infinite cubic well

In summary, the conversation discussed finding the stationary states of a particle in an infinite cubic well using the time independent Schrödinger equation and separation of variables. The question arose about the assumption that the variables X, Y, and Z cannot be zero, to which the response was that isolated zeros are acceptable for eigenfunctions.
  • #1
student111
16
0
Suppose one is to find the stationary states of a particle in an infinite cubic well. Inside the box the time independent SE is:

[tex] - \frac{\hbar}{2m} \big( \frac{\partial ^2 \psi}{\partial x ^2 } + \frac{\partial ^2 \psi}{\partial z ^2 } + \frac{\partial ^2 \psi}{\partial z ^2 } \big)= E\psi [/tex]

Using separation of variables: [tex] \psi = X(x)Y(x)Z(z) [/tex] we get:

[tex] YZ\frac{\partial ^2 X}{\partial x^2} + XZ\frac{\partial ^2 Y}{\partial y^2} + XY\frac{\partial ^2 Z}{\partial z^2} = \frac{-2mE}{\hbar ^2} XYZ [/tex]

After this one divides both sides by XYZ. My question is the following:
When dividing by XYZ one must assume that XYZ is different from zero. However the solution we obtain has lots of zeroes. Is this not a problem?

Thanks in advance
 
Last edited:
Physics news on Phys.org
  • #2
And?
 
  • #3
You just have to rule out XYZ being zero everhywhere.
At an isolated zero, the limit of the ratio X"/X is well behaved for an eigenfunction.
 

1. What is "Seperation of variables" in relation to the infinite cubic well problem?

The "Seperation of variables" method is a technique used to solve differential equations, such as those involved in the infinite cubic well problem. It involves separating a multi-variable equation into multiple single-variable equations, making it easier to solve.

2. What is the infinite cubic well problem?

The infinite cubic well problem is a popular example in quantum mechanics, where a particle is confined to a cubic potential well with infinitely high walls. This problem helps illustrate the concept of quantization, as the particle can only exist in discrete energy levels within the well.

3. How does the infinite cubic well problem relate to real-life systems?

The infinite cubic well problem may seem abstract, but it can actually be applied to real-life systems such as atoms and molecules. These systems can be represented as particles confined in a potential well, and the quantization of energy levels can help explain certain properties and behaviors.

4. What are the steps involved in solving the infinite cubic well problem using separation of variables?

The steps involved in solving the infinite cubic well problem using separation of variables include: setting up the Schrödinger equation for the system, separating the equation into multiple single-variable equations, applying boundary conditions to each equation, solving for the energy levels and corresponding wavefunctions, and checking for normalization of the wavefunctions.

5. Can the infinite cubic well problem be solved using other methods besides separation of variables?

Yes, the infinite cubic well problem can also be solved using other methods such as the variational method or the matrix diagonalization method. However, separation of variables is a commonly used and efficient method for solving this problem.

Similar threads

Replies
3
Views
283
  • Quantum Physics
2
Replies
56
Views
3K
Replies
29
Views
3K
  • Quantum Physics
Replies
11
Views
1K
  • Quantum Physics
Replies
19
Views
1K
Replies
14
Views
1K
  • Quantum Physics
Replies
17
Views
1K
Replies
1
Views
542
Replies
10
Views
1K
Replies
14
Views
1K
Back
Top