Calculate the energy of the five non interacting spin particles in a three-dimensional box of length L. (15)z²h? (a) 2mL? (25)z’h? (b) 2mL (24)z’n² (c) 2mL? (30)r’h (d) 2mL?
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Calculate the energy of the five non interacting spin 1/2 particles in a three dimensional box of length L.
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- A spin-particle is in the spin state |A), described by the ket 7 i |4) = 5√2 tu) +5√2 tu). (a) Verify that A) is normalised. (b) Using the spinor representation (+₂) = ₁ | +) = √/2₁ 11³) = ₁ 1 +2) = [8] find the values of c₁ and c₂ for which |A) = C₁|1₂) + C₂l+₂). (c) If the observable S₂ is measured in the spin state |A), what values can be obtained and what are their probabilities? (d) Find the expectation value of S₂ in the spin state [A). (e) With reference to the properties of angular momentum, explain briefly how the results of the Stern-Gerlach experiment provide evidence of the existence of spin.Explain what spin up and spin down is in quantum entanglement.I3
- Suppose you measure the angular momentum in the z-direction L, for an /= 2 hydrogen atom in the state | > 2 > |0 > +i/ |2 >. The eigenvalues of %3D V10 10 Lz are – 2h, -ħ, 0, ħ, 2ħfor the eigenvectors | – 2 >, |– 1>, |0 >, |1 >, |2 >, respectively. What is AL,? V31 10 7 19 25Rewrite S₁ S₂ in terms of S², |S₁|², 5₂|² by using the identity |S² = |S₁ + S₂|² = |S₁|² + |5₂|² +25₁ · 5₂. Use this to show that the combined spin angular momentum basis 5² for the electron and proton spins is an eigenstate basis for this dipole interaction.List all the reasons why a fourth quantum number (intrinsic spin) might have helped explain the complex optical spectra in the early 1920s
- What kind of particle would you expect to be made of the c baru configuration?What was used in the Stern-Gerlach experiment to creat an inhomogeneous magnetic field?Consider an electron in an external magnetic field in the SzS-direction, $\mathbf{B} B_z \hat{k}S. If the initial spin state of the electron is the eigenstate of SS_x$ with eigenvalue S+\hbar/2$, \begin{enumerate} \item[a)] Find the state of the system at time St$. \item [b)] Show that the system returns to its initial state and calculate the angular frequency. \item [c)] Calculates the expected value of SS_XS, SS_yS and SS_ZS as a function of time. \ end {enumerate}