Understanding the Adjoint Operator in Physics

• kuan
In summary, the adjoint operator is a unique linear operator that maps from one Hilbert space to another. It is defined by the orthonormality of bra and ket states, and it has the same eigenvalues as the operator it is adjoint to. Its physical interpretation is that it assists in representing the system mathematically and gives complex conjugate counterparts to state constants.
kuan
Dear All,

I am right now facing a problem of the adjoint operator.

From the mathematical point of view, if $$T$$ is a linear operator mapping from Hilbert space $$H$$ to another Hilbert space $$H'$$, there exist an unique adjoint operator $$T^{\dagger}$$ mapping from $$H'$$ back to $$H$$.

However I am right now struggling in understanding the physical meaning of the adjoint operator (not necessarily self-adjoint). The only thing I can think is that the operator $$T$$ and $$T^{\dagger}$$ have the same eigenvalues. Does this reveals some kind of symmetry? Could some one give me a bit hint for that?

Many thanks

Last edited:
For anyone in the future looking at this question, here's the main gist of the definition of an adjoint operator.
Assuming an orthonormal set of bra and ket states, if A is a an operator, then it will have the function

A = $\sum$i |bi> <ai|
so that when we operate on some state |aj>, we get the corresponding |bj>
A|aj> = |bj>
This is by definition of orthonormality, that <an|an>=1 and <an|am>=0 (same for b).

Meanwhile, for the very same set of states A is the exact definition, with the a and b reversed.
A = $\sum$i |ai> <bi|

As far as physical interpretation, well, we know that these bras and kets inform us of states of a system. When we observe the system (such as with the momentum operator -i$\hbar$$\frac{d}{dx}$), we may find particular eigenvalues and corresponding eigenfunction that serve as a basis of probable outcomes within the set of states. The adjoint operator serves the purpose of assisting in the mathematical representation of the scenario, giving complex conjugates counterparts to state constants.

I'm still a bit in the beginning of a grad course in Quantum I, if anyone can add this, by all means go for it. If I acquire more relevant info from the course/elsewhere later on I will share.

1. What is the adjoint operator in physics?

The adjoint operator, also known as the Hermitian conjugate, is a mathematical operation used to describe the relationship between linear operators in physics. It is denoted by a dagger symbol and is defined as the transpose of the complex conjugate of an operator.

2. How is the adjoint operator used in physics?

The adjoint operator is used to relate the inner product of two vectors in a Hilbert space. It is also used in quantum mechanics to describe the time evolution of a system and to calculate expectation values of observables.

3. What is the importance of the adjoint operator in physics?

The adjoint operator is important in physics because it allows us to perform calculations and make predictions about physical systems. It also helps us understand the symmetry and conservation laws of nature.

4. How is the adjoint operator related to the concept of Hermitian operators?

A Hermitian operator is one that is equal to its own adjoint. This means that the operator is self-adjoint and has real eigenvalues. In physics, Hermitian operators are associated with observables, which have real values in measurements.

5. Can the adjoint operator be applied to non-linear operators?

No, the adjoint operator can only be applied to linear operators. This is because the properties of linearity, such as superposition and homogeneity, are necessary for the adjoint operator to work properly. Non-linear operators do not satisfy these properties and therefore cannot have an adjoint.

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