- #1
ohhhnooo
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what is an eigenequation? what is the purpose of the eigenvalue? how does this fit into the schrodinger equation (particle in a box problem) ?
What is an Eigenequation and Eigenvalue in the Schrodinger Equation?
- Thread starter ohhhnooo
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In summary, the equation states that you have a system, represented by vector |Psi> and you want to measure the energy of it. You do this by applying an operator, H, and getting the eigenvalue, E.
- #2
Edgardo
- 706
- 17
An eigenequation is for example the following:
M x = b x
where M is a Matrix (for example a 3x3), x is a vector (3 components)
and b is a real number (could also be complex number).
You see that the Matrix doesn't change the direction of x, only it's length (right hand side of the equation).
x is called eigenvector and b eigenvalue of M.
Now in Quantum mechanics you have operators (instead of matrices)
and so called state vectors,
for example:
H |Psi> = E |Psi>
( M x = b x )
H is the Hamilton-Operator, |Psi> is your eigenvector and E the eigenvalue.
Whats the meaning of the equation above?
It just says that you got a system represented by the vector |Psi>
(for example electron in the Hydrogen atom).
And then you want to measure the energy. This is done by
'throwing' the operator H on your vector |Psi>. What comes out
is your eigenvalue E which is the energy.
Now what's the Schrödinger equation?
Suppose you want to examine the energy of the electron in the hydrogen atom. So you just apply H on |Psi> and get the energy E on the right hand side of the eigenequation.
The PROBLEM is, you don't know how your |Psi> looks like.
So here's where the SCHRÖDINGER equation comes into the play.
The Schrödinger equation is a differential equation,
which you have to solve in order to get your |Psi>. (solving the differential equation means you get a solution |Psi>)
You put your potential (square well potential for particle in a box, or Coloumb potential for hydrogen atom) into the Schrödinger equation and solve it. You get your |Psi> from it.
I hope I could help you.
-Edgardo
M x = b x
where M is a Matrix (for example a 3x3), x is a vector (3 components)
and b is a real number (could also be complex number).
You see that the Matrix doesn't change the direction of x, only it's length (right hand side of the equation).
x is called eigenvector and b eigenvalue of M.
Now in Quantum mechanics you have operators (instead of matrices)
and so called state vectors,
for example:
H |Psi> = E |Psi>
( M x = b x )
H is the Hamilton-Operator, |Psi> is your eigenvector and E the eigenvalue.
Whats the meaning of the equation above?
It just says that you got a system represented by the vector |Psi>
(for example electron in the Hydrogen atom).
And then you want to measure the energy. This is done by
'throwing' the operator H on your vector |Psi>. What comes out
is your eigenvalue E which is the energy.
Now what's the Schrödinger equation?
Suppose you want to examine the energy of the electron in the hydrogen atom. So you just apply H on |Psi> and get the energy E on the right hand side of the eigenequation.
The PROBLEM is, you don't know how your |Psi> looks like.
So here's where the SCHRÖDINGER equation comes into the play.
The Schrödinger equation is a differential equation,
which you have to solve in order to get your |Psi>. (solving the differential equation means you get a solution |Psi>)
You put your potential (square well potential for particle in a box, or Coloumb potential for hydrogen atom) into the Schrödinger equation and solve it. You get your |Psi> from it.
I hope I could help you.
-Edgardo
- #3
ohhhnooo
- 10
- 0
thanks alot!
FAQ: What is an Eigenequation and Eigenvalue in the Schrodinger Equation?
1. What is an eigenvalue and eigenequation?
An eigenvalue is a special type of scalar that represents the scaling factor of an eigenvector when multiplied by a linear transformation. An eigenequation is a mathematical equation that describes the relationship between an eigenvalue and its corresponding eigenvector.
2. What is the significance of eigenvalues and eigenvectors in linear algebra?
Eigenvalues and eigenvectors play a crucial role in linear algebra, as they are used to solve systems of linear equations, diagonalize matrices, and understand the behavior of linear transformations. They also have applications in fields such as physics, engineering, and computer science.
3. How do you find eigenvalues and eigenvectors?
To find eigenvalues and eigenvectors, you first need to set up and solve the eigenequation, which involves finding the determinant of a matrix and solving for its roots. Once you have the eigenvalues, you can find the corresponding eigenvectors by plugging in each eigenvalue into the original matrix and solving for the variables.
4. Can a matrix have multiple eigenvalues and eigenvectors?
Yes, a matrix can have multiple eigenvalues and eigenvectors. In fact, the number of eigenvalues and eigenvectors of a matrix is equal to its dimension. This means that a 3x3 matrix will have 3 eigenvalues and 3 corresponding eigenvectors.
5. What is the relationship between eigenvalues and determinants?
The eigenvalues of a matrix are equal to the roots of its characteristic polynomial, which is found by taking the determinant of the matrix and setting it equal to 0. This means that the determinant can be used to find the eigenvalues of a matrix, but it cannot give information about the eigenvectors.