In quantum mechanics, eigenstates of an operator are the states that the operator does not change. That is, the operator leaves the eigenstates “invariant.” Eigenstates are also called “eigenvectors.”
To find the eigenstates of an operator, we need to solve the eigenvalue equation:
O|ψ> = λ|ψ>
This equation says that the operator O acting on the state |ψ> gives us a new state that is a multiple (lambda, λ) of the original state |ψ>. So an eigenstate of O is a state that, when O acts on it, does not change.
The eigenvalue equation is a linear equation, so it has a unique solution for each eigenstate. That is, each eigenstate has a unique eigenvalue.
We can think of the eigenstates of an operator as the “direction” that the operator is pointing in. That is, if we have an operator that is a vector, then the eigenstates are the directions that the vector is pointing in.
The eigenstates of an operator form a complete basis for the space of states. That is, any state can be written as a linear combination of eigenstates of the operator.
The eigenstates of an operator are also the states that are “left unchanged” by the operator. That is, if we have an operator that is a matrix, then the eigenstates are the states that are not changed by the matrix.
The eigenstates of an operator are also the states that are “annihilated” by the operator. That is, if we have an operator that is a differential operator, then the eigenstates are the functions that are “killed” by the differential operator.
Other related questions:
Q: How do you find eigenstates of an operator?
A: There are many ways to find eigenstates of an operator. One way is to solve the eigenvalue equation for the operator.
Q: How do you find eigenstates of a Hamiltonian?
A: There are a few different ways to find eigenstates of a Hamiltonian. One way is to use the equation:
H|ψ> = E|ψ>
where H is the Hamiltonian, |ψ> is the eigenstate, and E is the eigenvalue.
Another way to find eigenstates is to diagonalize the Hamiltonian. This can be done by finding the eigenvectors and eigenvalues of the Hamiltonian matrix.
Q: What is eigenstates in quantum mechanics?
A: In quantum mechanics, an eigenstate is a state of a quantum system that is an eigenvector of the system’s Hamiltonian. This term is also used to refer to the corresponding eigenvalue.
Q: How do you find the eigenvalues of a Hermitian operator?
A: There are a number of ways to find the eigenvalues of a Hermitian operator. One way is to use the characteristic equation, which is given by:
det(H-λI)=0.
Another way is to use the spectral theorem, which states that the eigenvalues of a Hermitian operator are real and that the eigenvectors of a Hermitian operator form an orthonormal basis.
Bibliography
- Eigenvalues and eigenstates in quantum mechanics – YouTube
- Finding the eigenstates of an operator – Physics Stack Exchange
- 3.8: Eigenstates and Eigenvalues – Physics LibreTexts
- How to Find the Eigenvectors and Eigenvalues of an Operator
- Operator methods in quantum mechanics
- The Essentials of Quantum Mechanics – WUSTL Physics