Hermitian Operator: Definition & Overview

Click For Summary
SUMMARY

A Hermitian operator is defined as an operator in quantum mechanics (QM) that is equal to its own Hermitian conjugate. In matrix representation, this means that the diagonal elements (Mii) are real numbers, and the off-diagonal elements (Mij) are the complex conjugates of their transposed counterparts (Mji). Hermitian operators represent measurable quantities in QM, such as spin states, where the measurement of a z-directed spin by the operator \(\hat{S}_z\) on a spin-up electron state \(\left | + \right>\) yields a result of +\(\hbar/2\). A critical characteristic of Hermitian operators is that their eigenvalues are always real numbers.

PREREQUISITES
  • Understanding of quantum mechanics (QM) principles
  • Familiarity with linear algebra concepts, particularly matrices
  • Knowledge of complex numbers and their conjugates
  • Basic grasp of quantum state notation and operators
NEXT STEPS
  • Study the properties of Hermitian operators in quantum mechanics
  • Learn about eigenvalues and eigenvectors in the context of linear algebra
  • Explore the role of observables in quantum mechanics
  • Investigate the implications of measurement in quantum states
USEFUL FOR

Students and professionals in physics, particularly those focusing on quantum mechanics, as well as mathematicians interested in linear algebra applications in QM.

sm09
Messages
9
Reaction score
0
what is it?
 
Physics news on Phys.org
an operator that is hermitian!

in a matrix representation, it means that the diagonal Mii is real, and any Mij is the complex conjugate of Mji. this gives the hermitian conjugate of M (transpose and conjugate) is itself.or... google Hermitian

it represents a measurable quantity in QM.

ps.
am i right?
 
Last edited:
This question is well enough described in a any QM textbook. You do not make much effort to look there. In simple, Hermitian operator is the observable represetation, or if not rigorously speaking, it reflects the measurement procedure in a some quantum state. For example, let we have an spin-up directed electron state \left | + \right>. Measurement of the z-directed spin by \hat S_z in this state is reflected in the equality \hat S_z \left | + \right> = +\hbar/2 \left | + \right>. This mean the result you will get is +\hbar/2. A key feature of a Hermitian operator is real numbers of their eigenvalues.
 
Last edited:

Similar threads

Undergrad A strange definition for Hermitian operator
  • · Replies 3 ·
Replies
3
Views
1K
Undergrad Action of an Hermitian unitary operator member of SU(2)
  • · Replies 15 ·
Replies
15
Views
1K
High School Is the Momentum Operator Hermitian? A Proof
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad How to derive momentum operator in position basis via its definition?
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Completeness of Eigenfunctions of Hermitian Operators
  • · Replies 22 ·
Replies
22
Views
3K
Undergrad Motivation behind the Operator-Formalism in QM
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Eigenstates of particle with 1/2 spin (qbit)
  • · Replies 24 ·
Replies
24
Views
2K
Undergrad Proof of Commutative Operators for Simultaneous Measurement
  • · Replies 14 ·
Replies
14
Views
2K
Undergrad Dirac Notation for Operators: Ambiguity in Expectation Values?
  • · Replies 5 ·
Replies
5
Views
2K
Graduate Converting between field operators and harmonic oscillators
  • · Replies 2 ·
Replies
2
Views
1K