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Incorrect Question 9 0/1 pts The state % = to 4-1 + tzv. + tag 42...
2. (9 points total) Uncertainty relations. a) (1 point) Compute the commutator of the operators of...
2. (9 points total) Uncertainty relations. a) (1 point) Compute the commutator of the operators of coordinate and momentum in one dimension. b) (1 point) Two Hermitian operators A and B satisfy the relation [A, B] = iſ, where I is a number. Prove that I' is real. c) (1 point) Give the definition of the uncertainties A A and A B. d) (2 points) In this and subsequent parts of the question, we consider a normalized quantum stately) with...
3. (40 pts) A harronic oscillator has normalized energy eigenstates l e,) , n-0, 1, 2,...
3. (40 pts) A harronic oscillator has normalized energy eigenstates l e,) , n-0, 1, 2, The ladder operators have the properties and The momentum operator is given by |m/s 0)- (%)+2in)-in Suppose the oscillator is initially in the state a) (5 pts) What is the state |W() at later tiriest0 b) (5 pts) Write!ψ(1) as a column vector in the basis {1%), 1%), 19)} (5 pts) Write ()as a row vector in the basi),19) d) (5 pts) Construct the...
qm 2019.3 3. The Hamiltonian corresponding to the magnetic interaction of a spin 1/2 particle with...
qm 2019.3
3. The Hamiltonian corresponding to the magnetic interaction of a spin 1/2 particle with charge e and mass m in a magnetic field B is À eB B. Ŝ, m where Ŝ are the spin angular momentum operators. You should make use of expres- sions for the spin operators that are given at the end of the question. (i) Write down the energy eigenvalue equation for this particle in a field directed along the y axis, i.e. B...
4. 10 points The Spin operators for a spin-1/2 particle can be described by the Pauli...
4. 10 points The Spin operators for a spin-1/2 particle can be described by the Pauli matrices: 0 1 0 0 ,02= 0 -1 1 ¿ a) Write the normalized eigenvectors of Oz, I+) and 1-) which are defined such that 0z|+) = 1+) and 0z1-) = -1-), as column vectors in the same basis as the Pauli matrices given above. (You can assume without loss of generality that these eigenvectors are real.) (3 pts) b) Consider an eigenvector (V)...
Question #2: 6 pts] Find the eigenvalues and the normalized eigenvectors of the matrix 21 2...
Question #2: 6 pts] Find the eigenvalues and the normalized eigenvectors of the matrix 21 2 -1 2 Question #3: 10 pts] The electron in a hydrogen atom is a linear combination of eigenstates. Let us assume a limited linear combination to provide some sample calculations $(r, θ, φ) 2 ,1,0,0 + '2,1,0 (a) Normalize the above equation. (b) What are the possible results of individual measurements of energy, angular momentum, and the z-component of angular momentum? (c) What are...
0/1 pts Tect Question 9 Which of the following represents a rule for balancing a nuclear...
0/1 pts Tect Question 9 Which of the following represents a rule for balancing a nuclear equation? I. The total number of protons plus neutrons in the products and reactants must be the same II. The total number of each type of element in the products and reactants must be the same II. The total number of nuclear charges in the products and reactants must be the same IV.The total number of each type of elementary particle in the products...
Consider the state of a spin-1/2 particle 14) = v1o (31+z) + i] – z)) where...
Consider the state of a spin-1/2 particle 14) = v1o (31+z) + i] – z)) where | z) are the eigenstates of the operator of the spin z-component $z. 1. Show that [V) is properly normalized, i.e. (W14) = 1. 2. Calculate the probability that a measurement of $x = 6x yields 3. Calculate the expectation value (Šx) for the state 14) and its dispersion ASx = V(@z) – ($()2. 4. Assume that the spin is placed in the magnetic...
0/1 pts Incorrect Question 1 Which of the following is characteristic of a nuclear reaction? I. Electrons in atomic orb...
0/1 pts Incorrect Question 1 Which of the following is characteristic of a nuclear reaction? I. Electrons in atomic orbitals are involved in the breaking and forming of bonds II. Elements are converted from one to another III. Reaction rates are typically not affected by catalysts IV. Atoms are rearranged by the breaking and forming of chemical bonds I, II, and III II and III I and Il I, II, II, and IV lI and IV
Exercise 1: The helium atom and spin operators 26 pts (a) Show that the expectation value...
Exercise 1: The helium atom and spin operators 26 pts (a) Show that the expectation value of the Hamiltonian in the (sa)'(2a)' excited state of helium is given by E = $42.0) (Avo ) anordes ++f63,(-) (%13-12 r) 62(e)drz + løn.(r.) per 142, (ra)]" drų dr2 - / 01.(ru) . (ra) Anemia 02.(r.)61.(r.)dr; dr2 (1) Use the approximate, antisymmetrized triplet state wave function for the (Isa)'(280)' state as discussed in class. Hint: make use of the orthonormality of the hydrogenic...
rrect Question 11 0/1 pts Which of the following is characteristic of a nuclear reaction? 1....
rrect Question 11 0/1 pts Which of the following is characteristic of a nuclear reaction? 1. Electrons in atomic orbitals are involved in the breaking and forming of bonds II. Elements are converted from one to another III. Reaction rates are typically not affected by catalysts IV. Atoms are rearranged by the breaking and forming of chemical bonds III and IV I, II, III, and IV I, II, and III II and III I and II
2. (9 points total) Uncertainty relations. a) (1 point) Compute the commutator of the operators of coordinate and momentum in one dimension. b) (1 point) Two Hermitian operators A and B satisfy the relation [A, B] = iſ, where I is a number. Prove that I' is real. c) (1 point) Give the definition of the uncertainties A A and A B. d) (2 points) In this and subsequent parts of the question, we consider a normalized quantum stately) with...
3. (40 pts) A harronic oscillator has normalized energy eigenstates l e,) , n-0, 1, 2, The ladder operators have the properties and The momentum operator is given by |m/s 0)- (%)+2in)-in Suppose the oscillator is initially in the state a) (5 pts) What is the state |W() at later tiriest0 b) (5 pts) Write!ψ(1) as a column vector in the basis {1%), 1%), 19)} (5 pts) Write ()as a row vector in the basi),19) d) (5 pts) Construct the...
qm 2019.3
3. The Hamiltonian corresponding to the magnetic interaction of a spin 1/2 particle with charge e and mass m in a magnetic field B is À eB B. Ŝ, m where Ŝ are the spin angular momentum operators. You should make use of expres- sions for the spin operators that are given at the end of the question. (i) Write down the energy eigenvalue equation for this particle in a field directed along the y axis, i.e. B...
4. 10 points The Spin operators for a spin-1/2 particle can be described by the Pauli matrices: 0 1 0 0 ,02= 0 -1 1 ¿ a) Write the normalized eigenvectors of Oz, I+) and 1-) which are defined such that 0z|+) = 1+) and 0z1-) = -1-), as column vectors in the same basis as the Pauli matrices given above. (You can assume without loss of generality that these eigenvectors are real.) (3 pts) b) Consider an eigenvector (V)...
Question #2: 6 pts] Find the eigenvalues and the normalized eigenvectors of the matrix 21 2 -1 2 Question #3: 10 pts] The electron in a hydrogen atom is a linear combination of eigenstates. Let us assume a limited linear combination to provide some sample calculations $(r, θ, φ) 2 ,1,0,0 + '2,1,0 (a) Normalize the above equation. (b) What are the possible results of individual measurements of energy, angular momentum, and the z-component of angular momentum? (c) What are...
0/1 pts Tect Question 9 Which of the following represents a rule for balancing a nuclear equation? I. The total number of protons plus neutrons in the products and reactants must be the same II. The total number of each type of element in the products and reactants must be the same II. The total number of nuclear charges in the products and reactants must be the same IV.The total number of each type of elementary particle in the products...
Consider the state of a spin-1/2 particle 14) = v1o (31+z) + i] – z)) where | z) are the eigenstates of the operator of the spin z-component $z. 1. Show that [V) is properly normalized, i.e. (W14) = 1. 2. Calculate the probability that a measurement of $x = 6x yields 3. Calculate the expectation value (Šx) for the state 14) and its dispersion ASx = V(@z) – ($()2. 4. Assume that the spin is placed in the magnetic...
0/1 pts Incorrect Question 1 Which of the following is characteristic of a nuclear reaction? I. Electrons in atomic orbitals are involved in the breaking and forming of bonds II. Elements are converted from one to another III. Reaction rates are typically not affected by catalysts IV. Atoms are rearranged by the breaking and forming of chemical bonds I, II, and III II and III I and Il I, II, II, and IV lI and IV
Exercise 1: The helium atom and spin operators 26 pts (a) Show that the expectation value of the Hamiltonian in the (sa)'(2a)' excited state of helium is given by E = $42.0) (Avo ) anordes ++f63,(-) (%13-12 r) 62(e)drz + løn.(r.) per 142, (ra)]" drų dr2 - / 01.(ru) . (ra) Anemia 02.(r.)61.(r.)dr; dr2 (1) Use the approximate, antisymmetrized triplet state wave function for the (Isa)'(280)' state as discussed in class. Hint: make use of the orthonormality of the hydrogenic...
rrect Question 11 0/1 pts Which of the following is characteristic of a nuclear reaction? 1. Electrons in atomic orbitals are involved in the breaking and forming of bonds II. Elements are converted from one to another III. Reaction rates are typically not affected by catalysts IV. Atoms are rearranged by the breaking and forming of chemical bonds III and IV I, II, III, and IV I, II, and III II and III I and II
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