A three-state system has a complete orthonormal set of states |1⟩,|2⟩,|3⟩. With respect to this

step 1 Let us calculate the value of h bar first which is equal to hw bar multiplication 1, 0, 0, 0, mi

A three-state system has a complete orthonormal set of...

A three-state system has a complete orthonormal set of states $|1\rangle,|2\rangle,|3\rangle .$ With respect to this basis the operators $H$ and $B$ have matrices $$ H=\hbar \omega\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{array}\right) \quad B=b\left(\begin{array}{lll} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right) $$ where $\omega$ and $b$ are real constants. a. Are $H$ and $B$ Hermitian? b. Write down the eigenvalues of $H$ and find the eigenvalues of $B$. Solve for the eigenvectors of both $H$ and $B$. Explain why neither matrix uniquely specifies its eigenvectors. c. Show that $H$ and $B$ commute. Give a basis of eigenvectors common to $H$ and $B$.

A three-state system has a complete orthonormal set of states $|1\rangle,|2\rangle,|3\rangle .$ With respect to this basis the operators $H$ and $B$ have matrices
$$
H=\hbar \omega\left(\begin{array}{ccc}
1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & -1
\end{array}\right) \quad B=b\left(\begin{array}{lll}
1 & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0
\end{array}\right)
$$
where $\omega$ and $b$ are real constants.
a. Are $H$ and $B$ Hermitian?
b. Write down the eigenvalues of $H$ and find the eigenvalues of $B$. Solve for the eigenvectors of both $H$ and $B$. Explain why neither matrix uniquely specifies its eigenvectors.
c. Show that $H$ and $B$ commute. Give a basis of eigenvectors common to $H$ and $B$.

Key Concepts

Common Eigenbasis

A common eigenbasis is a basis comprised of vectors that are eigenvectors for two or more commuting operators. Constructing such a basis is key in simplifying problems and understanding the simultaneous properties of observables in a quantum system, as it allows the operators to be simultaneously diagonalized.

Commuting Operators

Commuting operators are those for which the order of application does not matter; their commutator is zero. This property is essential in quantum mechanics because it implies that the operators have a common set of eigenvectors, allowing for the simultaneous measurement of the related observables.

Hermitian Operators

Hermitian operators are those that equal their own adjoint; they have real eigenvalues and orthogonal eigenvectors. In quantum mechanics, they represent observable physical quantities because measurements yield real numbers. Moreover, Hermitian operators ensure that the dynamics or measurement predictions are consistent with the probabilistic interpretation of the theory.

Eigenvalues and Eigenvectors

The concept of eigenvalues and eigenvectors involves finding scalars and corresponding non-zero vectors such that when the operator acts on the vector, the result is the scalar times the same vector. These are central to understanding the spectrum of an operator in quantum mechanics, as they describe measurable outcomes and the state properties of the system.

Degeneracy

Degeneracy occurs when an operator has two or more linearly independent eigenvectors corresponding to the same eigenvalue. In such cases, the eigenvectors are not uniquely defined — any orthogonal linear combination of these eigenvectors is also an eigenvector. This non-uniqueness is an important concept when constructing bases in Hilbert spaces.

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