Classifying and Characterizing Linear Operators

Hermitian Operators: The Mathematics of Observables

In quantum mechanics, any measurable physical quantity—such as position, momentum, or energy—is called an observable and is represented by a linear operator. A core postulate of the theory, which we will formalize later, is that the result of any physical measurement must be a real number. This physical requirement imposes a strict mathematical constraint on the operators that can represent observables.

The expectation value (or mean value) of an observable $A$ for a system in a state $|\psi ⟩$ is given by $⟨A⟩=⟨\psi |A|\psi ⟩$. For this value to be real, it must be equal to its own complex conjugate: $⟨A⟩=⟨A{⟩}^{\ast }$. Let's see what this implies for the operator $A$.

Using the definition of the adjoint from our last lesson, where $(⟨\varphi |A|\psi ⟩{)}^{\ast }=⟨\psi |A^{†}|\varphi ⟩$, we can rewrite the right-hand side:

Therefore, the condition for a real expectation value becomes:

Since this must hold for any state $|\psi ⟩$ in the Hilbert space, the operators themselves must be equal. This leads to the defining property of an operator that can represent a physical observable:

An operator $A$ is Hermitian (or self-adjoint) if it is equal to its own adjoint:

The following video provides a clear and concise explanation of this motivation and introduces the key properties of Hermitian operators.

The properties introduced in the video are so fundamental that they are worth formalizing. The following text from the MIT OpenCourseWare notes for Quantum Theory I provides a rigorous statement and proof.

Let's briefly reconstruct the proofs for these two cornerstone properties, as they are excellent practice in manipulating bra-ket notation.

1. Eigenvalues of a Hermitian operator are real.
Let $A$ be a Hermitian operator with an eigenstate $|a⟩$ and corresponding eigenvalue $a$: $A|a⟩=a|a⟩$.
Taking the inner product from the left with $⟨a|$ gives:

Now, let's consider the complex conjugate of this equation:

Using the definition of the adjoint, the left side is $⟨a|A^{†}|a⟩$. Since $A$ is Hermitian, $A^{†}=A$, so we have:

Comparing our two main equations, we see that $a⟨a|a⟩=a^{\ast }⟨a|a⟩$. Since $|a⟩$ must be a non-zero vector, $⟨a|a⟩\ne 0$, which forces $a=a^{\ast }$. Thus, the eigenvalue $a$ must be real.

2. Eigenvectors of a Hermitian operator with distinct eigenvalues are orthogonal.
Consider two distinct eigenvalues $a_1\ne a_2$ and their corresponding eigenvectors:

Take the inner product of the first equation with $⟨a_2|$:

Now, take the adjoint of the second equation: $⟨a_2|A^{†}=a_2^{\ast }⟨a_2|$. Since $A$ is Hermitian and its eigenvalues are real, this becomes $⟨a_2|A=a_2⟨a_2|$. Taking the inner product of this with $|a_1⟩$ gives:

Comparing the two expressions for $⟨a_2|A|a_1⟩$, we get:

Since we assumed the eigenvalues are distinct ($a_1\ne a_2$), the inner product must be zero: $⟨a_2|a_1⟩=0$. The eigenvectors are orthogonal.

The third property, that the set of eigenvectors of a Hermitian operator forms a complete basis, is the statement of the Spectral Theorem. This is a profound result that we will explore in detail in the next lesson.

Unitary Operators: The Mathematics of Dynamics and Symmetries

While Hermitian operators describe static properties (observables), unitary operators describe dynamic processes and transformations that preserve the structure of the Hilbert space. This includes time evolution and symmetry operations like rotations and translations.

The fundamental physical principle here is the conservation of probability. If a state is normalized ($⟨\psi |\psi ⟩=1$), it must remain normalized after any physical transformation. More generally, the inner product between any two states must be invariant under such transformations, preserving their geometric relationship.

Let $U$ be an operator representing such a transformation. For any two states $|\psi ⟩$ and $|\varphi ⟩$, their transformed counterparts are $|{\psi }^{\prime }⟩=U|\psi ⟩$ and $|{\varphi }^{\prime }⟩=U|\varphi ⟩$. The invariance of the inner product requires:

Substituting the definitions of the transformed states:

Since this must hold for all $|\psi ⟩$ and $|\varphi ⟩$, we must have:

An operator satisfying this condition is called unitary. For finite-dimensional Hilbert spaces (and for bounded operators in infinite dimensions), this also implies $U{U}^{†}=I$, meaning the adjoint is the inverse:

The following reading provides a more rigorous derivation of this property and introduces the crucial link between unitary operators and Hermitian generators.

This connection between unitary operators and Hermitian generators is one of the most elegant and profound concepts in physics. It shows that for every continuous transformation (like time evolution or rotation), there is an associated observable (the generator) that is conserved if the system is symmetric under that transformation.

Key Properties of Unitary Operators:

1. Eigenvalues are complex numbers of unit modulus (pure phases).
   Let $U|u⟩=u|u⟩$. Then $⟨u|U^{†}=u^{\ast }⟨u|$. The inner product $⟨u|U^{†}U|u⟩$ is just $⟨u|u⟩$ because $U^{†}U=I$. But we can also calculate it as $(u^{\ast }⟨u|)(u|u⟩)=|u{|}^2⟨u|u⟩$. Equating these gives $|u{|}^2⟨u|u⟩=⟨u|u⟩$, which implies $|u{|}^2=1$.

2. They are generated by Hermitian operators.
   As shown in the reading, any continuous unitary transformation can be built from an infinitesimal one of the form $U(\delta ϵ)=I-iϵG$, where $G$ is a Hermitian operator called the generator.

   • Time evolution is described by the unitary operator $U(t)=e^{-iHt/\hslash }$. The generator is the Hamiltonian $H$, the observable corresponding to energy.
   • Spatial translation is described by $U(\mathbf{a})=e^{-i\mathbf{a}\cdot \mathbf{P}/\hslash }$. The generator is the momentum operator $\mathbf{P}$.
   • Rotation is described by $U(\mathbf{\theta })=e^{-i\mathbf{\theta }\cdot \mathbf{J}/\hslash }$. The generator is the angular momentum operator $\mathbf{J}$.

This structure, where symmetries (unitary operators) are linked to conserved quantities (their Hermitian generators), is a cornerstone of modern physics, from quantum mechanics to quantum field theory.

Projection Operators: The Mathematics of Measurement

The final class of operators we will study is central to the postulate of measurement. When we measure an observable, the state of the system is said to "collapse" onto the eigenspace corresponding to the measured eigenvalue. This "collapse" is mathematically described by a projection operator.

An operator $P$ is an orthogonal projection operator if it is both Hermitian and idempotent.

1. Hermitian: $P=P^{†}$
2. Idempotent: $P^2=P$

The idempotency condition means that applying the projection once has the same effect as applying it multiple times—once you've projected into a subspace, further projections into the same subspace do nothing new.

Key Properties of Projection Operators:

1. Eigenvalues are either 0 or 1.
   Let $P|\lambda ⟩=\lambda |\lambda ⟩$. Applying $P$ again gives $P^2|\lambda ⟩=P(\lambda |\lambda ⟩)={\lambda }^2|\lambda ⟩$.
   Since $P^2=P$, we have $\lambda |\lambda ⟩={\lambda }^2|\lambda ⟩$, which implies $\lambda ={\lambda }^2$. The only solutions are $\lambda =0$ and $\lambda =1$.

   • An eigenvalue of 1 corresponds to a vector that lies within the subspace onto which $P$ projects.
   • An eigenvalue of 0 corresponds to a vector that is orthogonal to that subspace.

2. They project onto subspaces.
   The simplest non-trivial projector is the one that projects onto a one-dimensional subspace spanned by a single normalized vector $|n⟩$. This operator is given by the outer product:

   $$P_n=|n⟩⟨n|$$

   Let's verify it's a projector:

   • Hermitian: $P_n^{†}=(|n⟩⟨n|{)}^{†}=|n⟩⟨n|=P_n$.
   • Idempotent: $P_n^2=(|n⟩⟨n|)(|n⟩⟨n|)=|n⟩(⟨n|n⟩)⟨n|=|n⟩(1)⟨n|=P_n$.

   The action of $P_n$ on an arbitrary state $|\psi ⟩$ is $P_n|\psi ⟩=|n⟩⟨n|\psi ⟩$. The result is a vector pointing in the direction of $|n⟩$ with a magnitude equal to the component of $|\psi ⟩$ along $|n⟩$.

Recall the completeness relation we derived in the last lesson: $I={\sum }_n|n⟩⟨n|$. This can now be reinterpreted as a decomposition of the identity operator into a sum of mutually orthogonal projection operators: $I={\sum }_n{P}_n$. This is a foundational concept we will build upon.

Conclusion

In this lesson, we have dissected the anatomy of linear operators in quantum mechanics, revealing how their structure is dictated by physical principles. Let's summarize the key takeaways:

• Hermitian Operators ($A=A^{†}$): They represent physical observables. Their defining property ensures that their expectation values are real. Their eigenvalues, which correspond to the possible outcomes of a measurement, are real, and their eigenvectors form a complete orthogonal basis for the state space.

• Unitary Operators ($U^{†}=U^{-1}$): They represent physical transformations that conserve probability, such as time evolution and symmetries. They preserve the inner product between states. Their eigenvalues are pure phases, and they are generated by Hermitian operators.

• Projection Operators ($P^2=P,P=P^{†}$): They represent the "collapse" of a state during a measurement. Their eigenvalues are 0 or 1, corresponding to states outside or inside the projection subspace, respectively.

These three classes of operators are the essential mathematical tools for formulating the postulates of quantum mechanics. In our next lesson, we will delve deeper into the Spectral Theorem, which provides a powerful way to decompose any Hermitian operator in terms of its eigenvalues and corresponding projection operators. This will unify the concepts we've discussed today and lay the final piece of mathematical groundwork before we state the full postulates of the theory.