Eigenvalue of lowering operator

In summary, the eigenvalue of a lowering operator is zero on the ground state, as defined by the state that gives zero when operated on by the lowering operator. This can easily be seen in the H.OSc. system, but it is important to note that the minimum state or vacuum state is not the same as the zero state. The proof can be found by researching Fock space. Coherent states are also important to consider, as they are eigenstates of the annihilation operator with a complex eigenvalue and behave classically in terms of minimizing uncertainty.
  • #1
izzmach
7
0
How to prove that eigenvalue of lowering operator is zero?
 
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  • #2
Eigenvalue of lowering operator is zero only on the ground state and this is so by the definition of ground state.
For other eigenvectors of the lowering operator(coherent states), eigenvalues are non-zero.
 
  • #3
Ground state or the absolute minimum state of the system is "defined" as the state which gives zero when operated by the
lowering operator.

It is easy to consider the H.OSc. and operate with the lowering operator till it gets to the min = h-bar omega/2 and if you apply furthe the lowering operator then you get zero. Of course the system maybe different in different situation and hence the minimum state may vary.

watch out that minimum state or the vacuum state is not the zero state, a zero state does not exists in nature.
The proof is kind of trival. google on Fock space and there you get your proof.
 
  • #4
Ladder operators (angular momentum or harmonic oscillators for example) act on states to go from one state to another state with an increased or decreased eigenvalue up to some normalization included. They are constructed so when you act on zero you get zero because that is the vacuum state by definition.

If you are talking about an eigenvalue of the lowering operator you must consider coherent states which are constructed to be eigenstates of the annihilation operator. The eigenvalue is some complex number. Coherent states are used in path integrals. If you look at the coherent states in the harmonic oscillator, they behave the most classically in terms of minimized uncertainty.
 

Related to Eigenvalue of lowering operator

1. What is the "eigenvalue" of a lowering operator?

The eigenvalue of a lowering operator is a number that corresponds to the amount by which the operator decreases the value of a given eigenvector. It is often denoted by the Greek letter lambda (λ).

2. How is the eigenvalue of a lowering operator calculated?

The eigenvalue of a lowering operator can be calculated by using the operator's corresponding matrix representation and applying linear algebra techniques, such as finding the roots of the characteristic polynomial.

3. What is the significance of the eigenvalue of a lowering operator?

The eigenvalue of a lowering operator is important because it provides information about the behavior of the operator on a given vector. It can also be used to determine the eigenvectors of the operator.

4. Can the eigenvalue of a lowering operator be negative?

Yes, the eigenvalue of a lowering operator can be negative. This indicates that the operator decreases the magnitude of the vector in addition to changing its direction.

5. How does the eigenvalue of a lowering operator relate to quantum mechanics?

In quantum mechanics, the eigenvalue of a lowering operator is used to represent the energy levels of a quantum system. The operator acts on the system's wavefunction, and its eigenvalue corresponds to the energy of the system in that particular state.

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