point (e). Use the result from the previous points, if
necessary.
- Quantum Mechanics I Need The Solution And Interpretation For Point E Use The Result From The Previous Points If Ne 1 (151.25 KiB) Viewed 22 times
[Quantum Mechanics] I need the solution and interpretation for point (e). Use the result from the previous points, if ne
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[Quantum Mechanics] I need the solution and interpretation for point (e). Use the result from the previous points, if ne
[Quantum Mechanics] I need the solution and interpretation for
point (e). Use the result from the previous points, if
necessary.
a Consider two identical, noninteracting spin-polarised particles with mass m and spin s in the following potential in one spatial dimension, So, if x € (0, L] , V (2) = 10, otherwise (a) We characterise the spin state of the two particles via the wave function (ms, ms) where ms, and ms, are the spin projection quantum numbers of the two particles. What does spin polarisation imply for the values of these two quantum numbers? Is the wave function x symmetric, antisymmetric or neither under particle exchange? Justify your answer. [3] (b) Write down the Hamiltonian Ĥ of the two-particle system. [4] Hint: We assume no interaction between spatial and spin degrees of freedom, hence spin can be entirely neglected in H. If you use abstract notation, indicate the tensor product structure of the operators in the two-particle Hilbert space HH. If you employ spatial representation, use the coordinates 21 and 22. (c) The eigenfunctions in spatial representation can be written as product states of spin and spatial degree of freedom, i.e., 0 (21,12, Mg,,ms) = V (11,22) x (ms, ms). For a given particle spin s, how does V (11,12) behave under particle exchange? Justify your answer. [5P] (d) For a given particle spins, determine an explicit expression for the ground-state spatial wave function 1,(1,12) and ground-state energy Eg. [8P] Hint: The single-particle eigenstates Un (*) in the potential V (2) are given by (n € N, n >0), 2 sin ("I"), if 2 € [0, L] , Un (2) = otherwise. (e) One can define the operator n., (1,2,...,řn) = 1 Dijitj 8(Ř – 7)8(R' – Fj) in spatial representation for a general N-particle system in d dimensions with the particle coordinates ñi Rd. Expectation values of this operator yield the joint probability density of finding one particle at position Ř and another particle at position R'. Determine the expectation value of this operator in the ground state of the problem discussed in this exercise for half-integer spin s. Interpret your result in view of the Pauli principle. [8P] = -{XX RR
point (e). Use the result from the previous points, if
necessary.
a Consider two identical, noninteracting spin-polarised particles with mass m and spin s in the following potential in one spatial dimension, So, if x € (0, L] , V (2) = 10, otherwise (a) We characterise the spin state of the two particles via the wave function (ms, ms) where ms, and ms, are the spin projection quantum numbers of the two particles. What does spin polarisation imply for the values of these two quantum numbers? Is the wave function x symmetric, antisymmetric or neither under particle exchange? Justify your answer. [3] (b) Write down the Hamiltonian Ĥ of the two-particle system. [4] Hint: We assume no interaction between spatial and spin degrees of freedom, hence spin can be entirely neglected in H. If you use abstract notation, indicate the tensor product structure of the operators in the two-particle Hilbert space HH. If you employ spatial representation, use the coordinates 21 and 22. (c) The eigenfunctions in spatial representation can be written as product states of spin and spatial degree of freedom, i.e., 0 (21,12, Mg,,ms) = V (11,22) x (ms, ms). For a given particle spin s, how does V (11,12) behave under particle exchange? Justify your answer. [5P] (d) For a given particle spins, determine an explicit expression for the ground-state spatial wave function 1,(1,12) and ground-state energy Eg. [8P] Hint: The single-particle eigenstates Un (*) in the potential V (2) are given by (n € N, n >0), 2 sin ("I"), if 2 € [0, L] , Un (2) = otherwise. (e) One can define the operator n., (1,2,...,řn) = 1 Dijitj 8(Ř – 7)8(R' – Fj) in spatial representation for a general N-particle system in d dimensions with the particle coordinates ñi Rd. Expectation values of this operator yield the joint probability density of finding one particle at position Ř and another particle at position R'. Determine the expectation value of this operator in the ground state of the problem discussed in this exercise for half-integer spin s. Interpret your result in view of the Pauli principle. [8P] = -{XX RR