Quantum Mechanical Operators

An operator is a symbol that tells you to do something to whatever follows that operator. They are commonly used to perform specified mathematical operations on certain functions. Operators may be used in mathematics, physics, or chemistry but their primary purpose is always to perform operations on variables. The quantum mechanical operators are used in quantum mechanics to operate on complex and theoretical formulations. The Hamiltonian operator is an example of operators used in complex quantum mechanical equations i.e. Schrodinger’s wave energy equation.

 is a function here, acting on a function (ψ). Now if  is an operator, it will map one state vector (ψ) into another one (Φ).

Operators in Quantum Mechanics

An operator can be imaginary as well as a complex quantity. It is a mathematical rule that acts upon a function and produces another function. As the operators of mathematical algebra, quantum mechanical operators also function under certain rules like addition, multiplication, etc. An operator has no meaning if it is written alone. Linear operators, SQR – Square operators, and Hermitian operators are some types of operators.

Classical observables have an associated quantum mechanical operator. In other words, for every measurable parameter in physical systems, there exists a quantum mechanical (QM) operator.

Why is there a need for QM operators?

In quantum mechanics, discrete particles are described as waves. These particles are well explained by classical mechanics, Newtonian mechanics. To shift the particle nature concept to wave alike requires the establishment of operators needed for the proper description and better understanding.

Classical mechanical quantities are represented as linear operators in the quantum field. Some such QM operators are listed below.

QM Operators

Some types of quantum mechanical operators (Linear and Hermitian) are discussed below.

Types of operators

In quantum mechanics, we deal with two types of operators.

Linear operators 

The operator is linear if it satisfies two conditions:

1. For the functions being added or subtracted, the function can be applied to all functions individually.
   • Â ( m + n ) = Âm + Ân
2. Constants are not affected by the application of linear operators.
   • Â (cm) = cÂm

Proof for Linear operators

• For sum and difference functions the operator can be applied to each function. For example, if P is an operator,

P (f + g)  =  Pf + Pg

Let,

P = d/dx

f = 3x²

g = 2x

d/dx (3x²+2x) = d/dx (3x²) + d/dx (2x)

6x + 2 = 6x + 2

• For multiplication condition,

P (af) = aPf

Let,

f = x²

P = d/dx

d/dx (ax²) = a.d/dx (x)²

a.2x = a.2x

Hermitian operator

The operator is said to be hermitian if it satisfies the following conditions. 

1. A hermitian operator can be flipped over to the other side. In other words, it justifies the complex conjugate transpose of matrices.
   • If  is hermitian, {g| .f} = {f| .g}
2. The eigenvalues of a hermitian operator are always real.
   • from above example, {f|Â .f} must be a real value.
3. The eigenvalues are orthonormal by convention for a hermitian operator. in other words, they have a complete set of orthonormal eigenfunctions (eigenvectors).

Proof for Hermitian operators

• If Ψ1 and Ψ2 are two functions and A is an operator then,

∫ Ψ1* (Â.Ψ2) . dτ = ∫ Ψ2 (Â.Ψ1*) . dτ

Let, 

Ψ1=e^-ix

Ψ2=cos(x)

Â=d²/dx²

the equation becomes,

∫ e^-ix(d²/dx² cos (x)) . dx = ∫ cos (x) (d²/dx² e^-ix) . dx

∫ e^-ix d/dx (-sinx) . dx = ∫ (cosx) / dx(-ie^-ix) . dx

–∫ e^-ix(cosx) . dx = –∫ cosx (ie^-ix) . dx

This proves that d²/dx² is a hermitian operator.

Key Takeaway(s)

Laplacian is also considered a quantum mechanical operator with a symbol of (∇²).

Observables and Operators in Quantum mechanics:

Observables:

• Position
• Momentum
• Kinetic energy
• Angular momentum
• Angular dipole moment, etc.

Operators:

• x (position)
• H (Hamiltonian)
• V (P.E)
• T (K.E)
• L (Angular momentum)
• ∇² (Laplacian operator), etc.

Concepts Berg

How many types of operators are there in quantum mechanics?

Operators used in quantum mechanics are:

1. Position [ x, y, z ]
2. Momentum [ p ]
3. Kinetic energy [ T ]
4. Potential energy [ V ]
5. Total Energy [ E ]
6. Hamiltonian [ H ]
7. Angular momentum [ L ]
8. Spin angular momentum [ S ]
9. Total angular momentum [ J ]
10. Transition dipole moment [ d ]

What is an energy operator in quantum mechanics?

Hamiltonian is the energy operator in quantum mechanics., It is applied in Schrodinger’s wave equation to get eigenvalues for energy.

What is the need for operators in quantum mechanics?

In quantum mechanics, just like mathematics, we are dealing with extreme-level superposition techniques. For complex equations to be solved and to get the values of parameters impossible otherwise, we use operators.

How are operators used in quantum mechanics?

An operator is applied to the quantum mechanical variables on the concept that, the eigenfunctions or eigenvectors e.g. ψ are regenerated along with eigenvalues after the implication of operator e.g. H. An example is H ψ  =  E ψ.

What are observables and operators in quantum mechanics?

Observable is a measurable quantity such as momentum, position, etc. Operators are generalized functions being applied to observables like hamiltonian is an operator for observable like energy.

What is a physical example of a unitary operator in quantum mechanics?

Operators whose inverse is their adjoint are termed, unitary operators or identity operators. They have the simplest example of rotations in R².

What is the physical meaning of dagger in quantum mechanics?

A dagger is a form of an operator used in corresponding to dual vectors. It specifies the transpose conjugate of any operator.

What does the Hermitian operator do in quantum physics?

Hermitian operators operate on functions that require real eigenvalues, interconvertability, and orthonormality.

What are the paradoxes in quantum mechanics?

Paradoxes are contradictory statements on any subject. In quantum mechanics, different answers than logical answers and negation of common sense relations are what paradoxes are in quantum mechanics.

What are the key experiments that established quantum mechanics?

The key experiments that made quantum mechanics include:

• Photoelectric effect
• Compton effect
• Annihilation of matter
• Young’s double-slit experiment, etc

What is math to quantum mechanics?

Mathematical formulations are the entire basis of quantum mechanics. This formalism uses mainly functional analysis like Hilbert space, etc.

What does the eigenvalue give us in quantum mechanics?

Eigenvalues give us the transformation parameter for any observable.

Where do the ladder operators for the orbital angular momentum eigenstates come from?

How to evaluate a Hamiltonian in Python (Quantum Mechanics, Band Structure)?

Hamiltonian can be evaluated in Python using the Finite difference method. The infinite difference method, eigenvalues, and eigenstates can be evaluated by diagnosing Hamiltonian.

What is the physical significance of Hamiltonian being a Hermitian?

Hamiltonian being a hermitian operator is the only reason it can normalize a wave function.

References

• Physical Chemistry | A molecular approach by Donald A. McQuarrie (University of California, Davis) and John D. Simon (DUKE University)
• A Textbook f Physical Chemistry | (Volume 1) by Mandeep Dalal (Maharshi Dayanand University, Rohtak, India)
• Operators in Quantum Mechanics (Hyperphysics, Department of Physics and Astronomy, Georgia State University)