Hermitian Operators Quiz

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Questions and Answers

What is true about the eigenvalues of a Harmitian operator?

All eigenvalues are real numbers.

Eigenvalues are not necessarily real. (correct)

Eigenvalues can be complex numbers.

Eigenvalues are always non-negative.

If two Harmitian operators A and B commute, which of the following is true?

Both A and B are not Harmitian.

A and B cannot have a common eigen function.

A must be equal to B.

There is also a Harmitian operator that follows the commutative law. (correct)

What does the condition (A.B - B.A)Ψ = 0 imply?

Ψ is not an eigen function of A or B.

Operators A and B are equal.

Operators A and B are NOT Harmitian.

Operators A and B commute with each other. (correct)

Which statement is correct regarding a linear operator and Harmitian operator?

Any power of a Harmitian operator is always a Harmitian operator. Signup and view all the answers

What can you conclude about two eigenstates Φα and Φα' of a Harmitian operator?

If they have different eigenvalues, they must be orthogonal. Signup and view all the answers

What does the expression Û = e^(iθA) define?

A unitary operator dependent on a Hermitian operator. Signup and view all the answers

In the context of the Harmitian operator, the notation [Â, â] = 0 signifies what?

The operator commutes with itself. Signup and view all the answers

What indicator is used to describe the simultaneous eigen function of operators A and B?

A simultaneous eigen function must exist for both A and B. Signup and view all the answers

What characterizes odd parity when using the parity operator?

Only one coordinate changes sign. Signup and view all the answers

What is the eigenvalue of the parity operator?

±1 Signup and view all the answers

Which of the following statements is true regarding the Heisenberg uncertainty principle?

The uncertainty in position and momentum is inversely related. Signup and view all the answers

How is the expectation value of an operator A represented in ket and bra notation?

<A> = <i|A|i> Signup and view all the answers

What needs to be satisfied for a function |Ψ> to be normalized?

<Ψ|Ψ> = 1 Signup and view all the answers

When proving that <Ψ|Φ> are orthogonal, what condition must be met?

<Ψ|Φ> must equal 0. Signup and view all the answers

What relationship holds true for the coefficients in the linear combination of states in example 3?

The sum of the squares of the coefficients must equal 1. Signup and view all the answers

Which of the following best describes the parity operator's commutativity?

It always commutes with Harmitian operators. Signup and view all the answers

What is the formula used to find the minimum radius r0 of the hydrogen atom?

r0 = (ħ^2 / mKe^2)^{1/3} Signup and view all the answers

What value does Emin take for the ground state energy of the hydrogen atom?

-13.6 eV Signup and view all the answers

Which equation represents the relationship for minimum energy in a harmonic oscillator?

Emin = ħw/2 Signup and view all the answers

At what condition does the minimum force dF/d(Δx) equal zero in a quantum harmonic oscillator?

Δx = Δx0 Signup and view all the answers

What is the significance of the Heisenberg uncertainty principle in the context of a hydrogen atom?

Electrons don't exist within the nuclei. Signup and view all the answers

What is the relationship between the uncertainties in measurements of two conjugate operators A and B, according to the uncertainty principle?

ΔA.ΔB ≥ 1/2 |<[C,A,B]>| Signup and view all the answers

When calculating the uncertainty in the measurement of operator A, which formula correctly represents this uncertainty?

(ΔA)^2 = <A^2> - (<A>)^2 Signup and view all the answers

What does the symbol ħ represent in the context of Heisenberg's uncertainty principle?

The reduced Planck's constant Signup and view all the answers

What is the minimum energy of an electron if it were to exist in the nucleus, based on the calculations provided?

20 MeV Signup and view all the answers

Which of the following statements correctly reflects the implications of the uncertainty principle for an electron's position and momentum?

It implies that both position and momentum cannot be measured with arbitrary precision simultaneously. Signup and view all the answers

Given Δx = 5×10^-15 m, what is the expression for the uncertainty in momentum Δpx according to Heisenberg's uncertainty principle?

Δpx = ħ/2 Δx Signup and view all the answers

In the derivation leading to the minimum value condition of function f(λ), what must be true for df(λ)/dλ?

df(λ)/dλ = 0 provides critical points in optimization Signup and view all the answers

What implication does the uncertainty principle have for electrons emitted during beta decay?

Their energy must always be less than 4 MeV. Signup and view all the answers

What is the natural width of the spectral line given by the equation ΔE?

$8.33 \times 10^{-8}$ eV Signup and view all the answers

Which equation represents the relationship between phase velocity and group velocity?

$dx/dt = \frac{Δ\omega}{Δk}$ Signup and view all the answers

What is the result of the commutation relation [x, px]?

iħ Signup and view all the answers

Which of the following identities represents the property of commutators involving sums?

[A,B+C] = [A,B] + [A,C] Signup and view all the answers

What is the result of the commutation relation [y, py]?

iħ Signup and view all the answers

If Δωt = Δkx, what can be inferred about the relationship between phase velocity and group velocity?

Phase velocity equals group velocity Signup and view all the answers

Which of the following statements about the operators x and px is correct?

[x, px] = iħ Signup and view all the answers

What does the equation [z, px] represent?

iħ ∂ψ/∂x Signup and view all the answers

How do you express the total angular momentum operator in terms of its components?

L = -iħ (y ∂/∂z - z ∂/∂y) Signup and view all the answers

What characterizes the relationship [Lx, Ly]?

iħLz Signup and view all the answers

Which property does the relation [Lx, Lz] possess?

iħLy Signup and view all the answers

What determines the relationship between two operators in the form [A,BC]?

[A,B]C + [A,C]B Signup and view all the answers

What is the derived equation for the relation [Lx, Lx]?

0 Signup and view all the answers

In quantum mechanics, what does the commutation relation [A,B] = -[B,A] imply?

The order of operation affects the outcome Signup and view all the answers

Study Notes

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.

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Description

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.