Question

Let |β1⟩ be an eigenvector of pˆ whose eigenvalue is zero. find ⟨α|β1⟩

Answer 1

In quantum mechanics, eigenvectors are associated with eigenvalues when an operator acts on them. For the given eigenvector |β1⟩ of the momentum operator with an eigenvalue of zero, the inner product ⟨α|β1⟩ will be zero as well.

Step-by-step explanation:

In quantum mechanics, eigenvectors are associated with eigenvalues when an operator acts on them. In this case, let's consider the eigenvector |β1⟩ of the operator p-hat (momentum operator) with an eigenvalue of zero. The expression we need to find is ⟨α|β1⟩, which represents the inner product between the eigenvector |β1⟩ and another vector |α⟩.

The inner product between two vectors |α⟩ and |β⟩ is given by ⟨α|β⟩ = ∑i (αi)(βi)*, where * denotes the complex conjugate. Since the eigenvalue of |β1⟩ is zero, we can conclude that |β1⟩ is a null vector, meaning all its components are zero. Therefore, ⟨α|β1⟩ will also equal zero.