angular momentum MQC

Angular Momentum

1. Orbital Angular Momentum Operator


  • Operator in Schrödinger Representation: The orbital angular momentum operator in spherical coordinates is represented using partial derivatives and spherical harmonics.

    L = -i\hbar\left(\hat{e}\theta \frac{\partial}{\partial \phi} + \hat{e}\phi \frac{\partial}{\partial \theta}\right)

  • Conservation of L: Angular momentum is conserved in a central force field.

2. General Properties

  • Vector Form:

    • Angular momentum (L\mathbf{L}) is defined as L=r×p\mathbf{L} = \mathbf{r} \times \mathbf{p}, where p\mathbf{p} is linear momentum.

    • Magnitude of angular momentum is conserved in closed systems.

  • Heisenberg Uncertainty Principle (HUP):

    • The uncertainty in position (Δz\Delta z) and momentum (Δpz\Delta p_z) cannot be simultaneously well-defined.

    • ΔzΔpz≥ℏ2\Delta z \Delta p_z \geq \frac{\hbar}{2}.

    • When one quantity is precisely defined, the uncertainty in the other increases.

3. Quantum Aspects of Angular Momentum

  • Commutation Relations:

    • For angular momentum components $(L_x, L_y, L_z)$:

    • [Li,Lj]=iℏϵijkLk[L_i, L_j] = i\hbar \epsilon_{ijk} L_k, showing they cannot have common eigenstates.

  • Central Forces and Angular Momentum:

    • In the presence of a central force, angular momentum is conserved.

4. Measurement and Eigenstates

  • Eigenvalue Interpretation:

    • The possible measurements of angular momentum are quantized.

    • Possible eigenvalues for $L^2$ and $L_z$ depend on quantum numbers $l$ and $m_l$:

    • L2∣l,ml⟩=ℏ2l(l+1)∣l,ml⟩L^2 |l, m_l\rangle = \hbar^2 l(l+1)|l, m_l\rangle, Lz∣l,ml⟩=ℏml∣l,ml⟩L_z |l, m_l\rangle = \hbar m_l |l, m_l\rangle.

5. Spin Angular Momentum

  • Definition:

    • Spin is an intrinsic property of particles exhibited as an internal angular momentum.

    • Represented by operator S^\hat{S}, similar commutation relations to $L$ exist.

  • HUP in Spin:

    • Similar uncertainty constraints apply, with the possibility of total angular momentum being quantized.

6. Combined Systems and Quantum States

  • Quantum States:

    • A state can be an eigenvector for both $L^2$ and $S^2$, with common quantization rules.

  • Direct Sum and Tensor Product:

    • The combined state of a system can be expressed via direct sums and tensor products of angular momentum states.

    • E=E1⊕E2E = E_1 \oplus E_2 for vector space representation where interactions occur in combined eigenstates.

7. Conclusion

  • Precession and Uncertainty:

    • The behavior of angular momentum in quantum systems exhibits classical counterparts but encompasses uncertainties defined by HUP.

    • Angular momentum can be defined as a vector field precessing around an axis, quantified inherently in quantum mechanics.