Hermitian Operators Quiz

Eigenvalues of Hermitian Operators

• The eigenvalues of Hermitian Operators are always real.
• Hermitian operators follow the orthogonality principle, meaning that eigenstates corresponding to distinct eigenvalues are orthogonal.

Commuting Hermitian Operators

• If two Hermitian Operators commute, there exists a common eigenfunction, or there exists a Hermitian operator that is also a simultaneous eigenfunction of both operators.
• For two Hermitian operators A and B to commute, the condition AB - BA = 0 must be true.

Properties of Hermitian Operators

• Hermitian operators are linear operators, and they follow the commutative law.
• Any power of a Hermitian operator is also Hermitian.
• Every operator commutes with itself.
• Linear combinations of Hermitian operators are also hermitian.
• Each Hermitian operator can be used to generate a unitary operator using the formula Û=e^(iθA), where θ is a real number.

Simultaneously Eigen Functions

• A function Ψ is a simultaneously eigenfunction of two operators A and B if it satisfies the following conditions:
  • A Ψ = aΨ
  • B Ψ = bΨ
  • Â(B̂Ψ) = B̂(ÂΨ) = a(B̂Ψ) = abΨ and Â(B̂Ψ) = B̂(ÂΨ) = b(ÂΨ) = abΨ
  • This means that the operators A and B commute [ÂB̂]= 0.
• If two operators commute, they have common eigenfunctions.

Parity Operator

• The parity operator (or space inversion operator) inverts the sign of space coordinates when applied to a function.
• It has eigenvalues of ±1.
• Even parity occurs when the sign of the coordinate is not changed by the Parity Operator.
• Odd parity occurs when the sign of at least one coordinate is changed by the Parity Operator.

Properties of the Parity Operator

• Eigenvalue ±1
• Linear Operator
• Hermitian Operator
• It always commutes with Hermitian operators
• It represents a constant motion

Ket and Bra Notation

• Ket Notation |i> represents an initial state.
• Bra Notation <i| represents a final state.
• Bra and Ket are complex conjugates of each other.
• Replacing Ket with Bra: <Ψ| = (|Ψ>)***

Example 1:

• The example demonstrates how to calculate <Ψ| and <Φ| given ket vectors |Ψ> and |Φ>.
• It also shows how to prove orthogonality between two vectors using the inner product, which is zero for orthogonal vectors.

Example 2:

• This example shows how to normalize a ket vector |Ψ> by applying a normalization constant a to it.
• It also shows how to calculate the normalization constant to adjust the vector's length to one.

Example 3:

• This example shows how to find the value of t in a linear combination of ket vectors.
• It emphasizes that the sum of squares of the coefficients in a linear combination of normalized ket vectors must equal one.
• In this case, t is one of the coefficients, and its value can be found by solving the equation t^2 + (1/2)^2 + (1/4)^2 = 1.

Heisenberg's Uncertainty Principle

• This principle states that the product of uncertainties in position Δx and momentum Δpx of a particle is at least half of Planck's constant ħ/2.
• The uncertainty principle applies to any pair of canonically conjugate variables.
• Mathematically, this can be expressed as: Δx.Δpx = h.

Application of the Uncertainty Principle

• Non-existence of electrons in the nucleus: By applying the position uncertainty of the atomic nucleus and the uncertainty of momentum, a minimum energy for the election is obtained.
• This result is greater than the energy observed in beta decay, indicating that electrons cannot exist stably within the nucleus.
• Ground State Energy of the Hydrogen Atom: Using uncertainty principle, the ground-state energy of hydrogen can be estimated, which is close to the value obtained from a more rigorous calculation.
• Ground State Energy of a Harmonic Oscillator: Utilizing uncertainty principle, the minimum energy of a harmonic oscillator is estimated, which coincides with the ground state energy of the oscillator.
• Natural Width of Spectral Line: The uncertainty principle explains the broadening of spectral lines due to the uncertainty in the energy of the states involved in the transitions.
• Phase and Group Velocity: The concepts of phase velocity and group velocity are introduced and linked to the uncertainty principle in the context of wave superposition.

Commutator Algebra

• This section outlines some important properties of the commutator, a mathematical operation that measures the extent to which two operators fail to commute.

Some Operators

• This section presents several important operators in quantum mechanics, including position operators x, y, z and momentum operators px, py, pz.
• It provides a worked example to calculate the commutator between the position operator x and the momentum operator px as well as others.

Communication Relations

• This section presents crucial commutator relations involving observable operators.
• It also includes examples of calculating the commutator of differentoperators, including the position, momentum, and angular momentum operators, and shows how these relations are derived using the fundamental commutation relations.

Test your knowledge on Hermitian operators, their eigenvalues, and the properties that define them. This quiz covers concepts such as orthogonality, commutation, and linear combinations of Hermitian operators. Ideal for students studying advanced mathematics or quantum mechanics.