Quantum Mechanics: Spin Particles Quiz
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Questions and Answers

For a spin 1/2 particle, what does the spinor t represent?

  • A combination of spin-up and spin-down states
  • Orbital angular momentum state
  • Spin-down state
  • Spin-up state (correct)

    • The energy levels of a spin 3/2 particle, as described in the example, are degenerate.

    False (B)

    In the Hamiltonian provided, what are the constants denoted by?

    Greek letters : and ;

    The Hamiltonian is diagonal in the ______ basis.

    <p>s m</p> Signup and view all the answers

    Match the following to spin operators using the example calculations:

    <p>: = Energy scaling constant related to spin ; = Energy scaling constant related to spin Sx = Spin operator along the x-axis Sy = Spin operator along the y-axis Sz = Spin operator along the z-axis</p> Signup and view all the answers

    In the expression for the energy levels of the spin 3/2 particle, what quantum number does 'm' represent?

    <p>Spin magnetic quantum number (C)</p> Signup and view all the answers

    The total wave function for a spin-down particle is represented by the spinor u.

    <p>True (A)</p> Signup and view all the answers

    What are the four values that m can take for a particle with spin s = 3/2?

    <p>-3/2, -1/2, 1/2, 3/2</p> Signup and view all the answers

    What does the orthonormality condition for angular momentum eigenstates state?

    <p>The eigenstates corresponding to different angular momenta are orthogonal. (A)</p> Signup and view all the answers

    The state |j, m> is an eigenstate of J+.

    <p>False (B)</p> Signup and view all the answers

    What is the value of the expectation value of Jx in the state |j, m>?

    <p>0</p> Signup and view all the answers

    The operator J- is known as the ______ operator.

    <p>lowering</p> Signup and view all the answers

    Match the angular momentum operators with their effect on the state |j, m>:

    <p>J+ = Raises the m value J- = Lowers the m value J^2 = Gives the total angular momentum eigenvalue Jz = Gives the angular momentum in the z-direction</p> Signup and view all the answers

    According to the content, what is J+ |j, m> equal to?

    <p>$h\sqrt{j(j+1) - m(m+1)} |j, m+1&gt;$ (B)</p> Signup and view all the answers

    The expectation values <Jx^2> and <Jy^2> are not equal in the state |j, m>.

    <p>False (B)</p> Signup and view all the answers

    What is the expression for J- |j, m>?

    <p>$h\sqrt{j(j+1) - m(m-1)} |j, m-1&gt;$</p> Signup and view all the answers

    The expectation value of J_y in the state |j, m> is equal to ______.

    <p>0</p> Signup and view all the answers

    What is the expression for the expectation value of <Jx^2>?

    <p>$(h^2/2) [j(j+1) - m^2]$ (A)</p> Signup and view all the answers

    In the eigenbasis of $|j, m\rangle$, which operators are represented by diagonal matrices?

    <p>J^2 and Jz (B)</p> Signup and view all the answers

    The operators $J_+$ and $J_z$ commute with each other.

    <p>False (B)</p> Signup and view all the answers

    If $j=1$, what are the possible values of m?

    <p>-1, 0, 1</p> Signup and view all the answers

    The diagonal elements of the matrix representing $J^2$ are equal to $\hbar^2 j(j+1)$ and the diagonal elements of the matrix representing $Jz$ are equal to $\hbar$ times ______.

    <p>m</p> Signup and view all the answers

    Match the angular momentum operators with their matrix representations in the |j, m⟩ basis:

    <p>$J^2$ = Diagonal $J_z$ = Diagonal $J_+$ = Non-Diagonal $J_x$ = Non-Diagonal $J_y$ = Non-Diagonal</p> Signup and view all the answers

    What is the result of the operator $J_z$ acting on the eigenstate $|j, m\rangle$?

    <p>$\hbar m |j, m\rangle$ (B)</p> Signup and view all the answers

    The matrices of Jx and Jy are diagonal in the |j, m> eigenbasis.

    <p>False (B)</p> Signup and view all the answers

    What is the value of J_z^3 for j=1?

    <p>h̄²Jz</p> Signup and view all the answers

    What happens when the operator J$^{+}$ acts on the state $|j,m\rangle$?

    <p>It raises the second quantum number m by one unit. (B)</p> Signup and view all the answers

    The eigenvalue of J$^{2}$ is always negative.

    <p>False (B)</p> Signup and view all the answers

    What is the significance of the upper limit for the quantum number m?

    <p>It indicates a maximum state which cannot be raised further.</p> Signup and view all the answers

    The relation J$^{+}$|j,m$ angle$ is proportional to |j,m+1$ angle, which can be expressed as J$^{+}$|j,m$ angle = ______|j,m+1 angle.

    <p>C_{j,m}</p> Signup and view all the answers

    Match the operators with their corresponding effects on the quantum states:

    <p>J$^{+}$ = Raises the second quantum number m by one unit J$^{-}$ = Lowers the second quantum number m by one unit J$^2$ = Determines the eigenvalue relation J$_{z}$ = Acts on defining the z-component of angular momentum</p> Signup and view all the answers

    What is the first step in determining the eigenfunctions of L according to the text?

    <p>Applying L to the eigenfunctions Ylm. (D)</p> Signup and view all the answers

    The Legendre differential equation includes a term that is dependent on .

    <p>False (B)</p> Signup and view all the answers

    What mathematical functions are the solutions to the Legendre differential equation expressed in terms of?

    <p>associated Legendre functions</p> Signup and view all the answers

    The equation for $lm(A)$ is given by $lm(A) = Clm P$^{}$(cos A)$, where $P$^{}$ represents the associated Legendre function.

    <p>lm</p> Signup and view all the answers

    What is the relationship between $P_l^m(x)$ and $P_l^{-m}(x)$?

    <p>$P_l^{-m}(x) = P_l^m(x)$ (A)</p> Signup and view all the answers

    The Legendre polynomial, $P_l(x)$, is defined by the ______ formula.

    <p>Rodrigues</p> Signup and view all the answers

    Match the following terms with their description:

    <p>Ylm = Eigenfunctions of L Plm(cos A) = Associated Legendre functions Pl(x) = Legendre polynomial L = Angular momentum operator</p> Signup and view all the answers

    The associated Legendre functions $P^{lm}(x)$ are derived by taking m derivatives of the Legendre polynomial $P_l(x)$.

    <p>True (A)</p> Signup and view all the answers

    Which of the following expressions is equal to $N J_x^2 O$?

    <p>$\frac{1}{4} N (J_+ + J_-)^2 O$ (D)</p> Signup and view all the answers

    The expectation value of $J_x$ is always non-zero.

    <p>False (B)</p> Signup and view all the answers

    What is the relationship between $N J_x^2 O$ and $N J_y^2 O$?

    <p>$N J_x^2 O = N J_y^2 O$</p> Signup and view all the answers

    The uncertainty in $J_x$ multiplied by the uncertainty in $J_y$, $\sigma_{J_x}\sigma_{J_y}$, is proportional to ______.

    <p>$h^2$</p> Signup and view all the answers

    Given that $N J_+^2 O = N J_-^2 O = 0$, what is the value of $N J_x^2 O$?

    <p>$\frac{1}{4} N (J_+ J_- + J_- J_+) O$ (D)</p> Signup and view all the answers

    The expression $j(j+1)-m^2$ is always positive.

    <p>False (B)</p> Signup and view all the answers

    What is the expression for $N J_x^2 O + N J_y^2 O$ in terms of $N J^2 O$ and $N J_z^2 O$?

    <p>$N J^2 O - N J_z^2 O$</p> Signup and view all the answers

    The expectation value $N J_z O$ is equal to ______$h\hbar$ when acting on the state $|jm\rangle$.

    <p>m</p> Signup and view all the answers

    Match the following operators with their corresponding actions:

    <p>$J_+$ = Raises the angular momentum state $J_-$ = Lowers the angular momentum state $J_x$ = Angular momentum projection on the x-axis $J_y$ = Angular momentum projection on the y-axis</p> Signup and view all the answers

    Given that $N J_x O = 0$ and $N J_y O = 0$, what quantity does $\sigma_{J_x} \sigma_{J_y}$ simplify to?

    <p>$\sqrt{N J_x^2 O N J_y^2 O}$ (C)</p> Signup and view all the answers

    Study Notes

    Chapter 5: Angular Momentum

    • This chapter introduces three-dimensional problems, a crucial prelude to Chapter 6, essential for understanding atomic systems.
    • Angular momentum is vital in both classical and quantum mechanics, particularly for systems with spherically symmetric potentials.
    • Bohr's model of the hydrogen atom uses quantized angular momentum.
    • Quantum angular momentum is fundamental to understanding molecular, atomic, and nuclear systems.
    • Classical angular momentum (L) is defined as the cross product of position (r) and momentum (p).
    • The orbital angular momentum operator (L) is obtained by substituting position (R) and momentum (P = -iħ∇) operators into the classical expression.
    • The Cartesian components of the angular momentum operator Lx, Ly, and Lz are given by corresponding equations.
    • The square of the angular momentum (L²) is a conserved quantity that commutes with the Cartesian components of the angular momentum.
    • Commutation relations between angular momentum components are crucial; they involve ih and one of the other components.
    • Angular momentum operators do not commute with each other.
    • Simultaneous measurement of non-commuting operators is not possible due to the uncertainty principle.
    • The commutation relations allow the derivation of the general formalism of angular momentum, which is independent of a particular representation.
    • Key example of commutators are calculated
    • A general angular momentum operator (J) is defined, with commutation relations similar to those of orbital angular momentum.
    • The square of the angular momentum operator (J²) is a scalar operator, commuting with its Cartesian components.
    • The key is to diagonalize both the square and one of the components (often Lz).
    • Eigenstates of the angular momentum operator (J²) and one component (e.g., Jz) are introduced.
    • The eigenvalues of J² and Jz are directly related to the quantum number j and m, with m ranging from -j to +j, where j can be integer or half-integer.
    • The eigenvalues of J² are given by ħ²j(j + 1) and
    • The quantization of angular momentum determines the possible values of j and m.
    • Raising and lowering operators (J+, J−) are introduced and their role in manipulating the eigenvectors is discussed.
    • The expectation values of the x and y components of the angular momentum operator are zero.

    General Formalism of Angular Momentum

    • Commutation relations are introduced defining the angular momentum operators.
    • Eigenstates and eigenvalues of the square of angular momentum and components.
    • The factors ħ and the importance of orthonormality of the obtained eigenstates are stated.
    • Raising and lowering operators (J+ and J−) manipulate eigenstates, affecting only the eigenvalue m while not changing the eigenvalue a.
    • The quantum number m ranges from –j to +j
    • The quantum number j is either an integer or a half-integer.
    • The square of the angular momentum (J²) has eigenvalues ħ²j(j + 1).

    Matrix Representation of Angular Momentum

    • The general formalism is extended to matrix representations in a given basis of orthonormal eigenstates |j,m).
    • Specific examples for j=1 are given.
    • Matrix representations of operators Lx, Ly, and Lz are shown with explicit forms.
    • The results for operators are derived; the form of matrices is important.

    Geometrical Representation of Angular Momentum

    • Angular momentum represented as a rotating vector on the surface of a cone around its z axis.

    Experimental Evidence of Spin

    • Stern-Gerlach experiment demonstrated electron's intrinsic angular momentum (spin).
    • Spin's direction is quantized, unlike orbital angular momentum.
    • Spin is an intrinsic property of a particle.

    Spin Angular Momentum

    • Spin has a quantum mechanical origin.
    • The spin angular momentum operator (Ŝ) has properties analogous to orbital angular momentum operators
    • The components (Sx, Sy, Sz) of the spin angular momentum operator follow analogous commutation relations to those for orbital angular momentum.
    • Spin has eigenvalues related to a quantum number s, with s= 1/2 for an electron.
    • The Stern-Gerlach experiment demonstrated the quantization of electron spin.
    • Electron spin plays a key role in atomic and molecular systems.

    Eigenfunctions of Orbital Angular Momentum

    • The eigenfunctions of the orbital angular momentum operators are spherical harmonics(Yl,m(θ, φ)).
    • The functions depend only on the angular variables θ and φ.
    • The spherical harmonics are derived using differential equations.
    • The orthogonality and completeness of spherical harmonics are mentioned.

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    Description

    Test your knowledge on spin 1/2 and spin 3/2 particles in quantum mechanics. This quiz covers topics such as spinors, Hamiltonians, and angular momentum operators. Challenge yourself with questions about eigenstates and expectation values.

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