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An electron is in the spin state x= A () (a) Determine the normalization constant A....
1-3i An electron is in the spin state x A a. Determine the normalization constant A...
1-3i An electron is in the spin state x A a. Determine the normalization constant A b. Find the expectation value for S c. Find the expectation values for Sx d. Find the expectation values for Sz e. If you measure S on this electron, what values you get, and what is the probability of cachn? f. If you measure on this electron, what values you get, and what is the probability of cachn? g. If you measure Sy on...
- A() An electron is in the spin state x= A Determine the normalisation constant 1...
- A() An electron is in the spin state x= A Determine the normalisation constant 1 i A, and then find the expectation value of the z-component of the spin operator h [6]
4. Spin (10 marks) Suppose an electron is in a state such that its spin can...
4. Spin (10 marks) Suppose an electron is in a state such that its spin can be described by a linear superposition of the eigenspinors of S +A 32 2/22 (a) Normalise the state. (b) What are the possible outcomes of a measurement of the z-component of the spin? What is the probability of each possible result? (c) What are the expectation value and uncertainty of the z-component of the spin? (d) What are the possible results of measuring the...
3 of 5 Problem #2 (show all your work!) spin For the state +), calculate the...
3 of 5 Problem #2 (show all your work!) spin For the state +), calculate the expectation values and uncertainties for measurements of S, Sy, and S,
4. A (one dimensional) particle in a box of length 2a (i.e., zero potential energy) is...
4. A (one dimensional) particle in a box of length 2a (i.e., zero potential energy) is represented by the wavefunction v(x) 0, otherwise a. Sketch the wavefunction. Write down the (time independent) Schrodinger equation. Show whether or not the wavefunction is a solution to the equation. b. What does it mean physically if the wavefunction of the particle is NOT a solution to the Schrodinger equation? Explain. c. Determine the normalization constant A. 5. Same system. Find the average or...
3- A one-dimensional harmonic oscillator wave function is ψ(x) = Axe-bx2 a) Find the total energy...
3- A one-dimensional harmonic oscillator wave function is ψ(x) = Axe-bx2 a) Find the total energy E b) Find the constant b c) Find the normalization constant A. d) Find the expectation value of x, e) Find the uncertainty in x, Ох. f) Find the expectation value of p g) Find the uncertainty in p, Op For the Hamiltonian matrix shown below:
3- A one-dimensional harmonic oscillator wave function is ψ(x) = Axe-bx2 a) Find the total energy E b)...
b 2. Suppose a spin-2 particle is in the state that particle. 2,0) + 2,1) Find...
b 2. Suppose a spin-2 particle is in the state that particle. 2,0) + 2,1) Find the expectation value of S, for D 3. In the t spin-1/2 basis, consider the two operators 2 1 12d B- (2 i A= ni 2 (a) Find the commutator [A, B (b) Suppose we measure a number of particles in state |t), using A and B. Find the average values (A) and (B) from these measurements. (c) Use the uncertainty principle to find...
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...
lsa(1) lsB(1) 1Isa(2) 1sja 7. Consider this two-electron wave function: ψ-C Write the expression for ψ that comes from expanding the determinant. Find the normalization constant, C. The 1s orbitals a...
lsa(1) lsB(1) 1Isa(2) 1sja 7. Consider this two-electron wave function: ψ-C Write the expression for ψ that comes from expanding the determinant. Find the normalization constant, C. The 1s orbitals are orthonormal, and so are the spin orbitals. a) b) Using your answer from (a), show that the wave function factors into a spin part and a spatial part. Hint: It may help to rewrite each spin orbit so its spatial and spin factors are clearer. For instance, rewrite Isa...
ANSWER ALL QUESTIONS 1. (a) The hydrogen atom wave functions are written as Unim. State the...
ANSWER ALL QUESTIONS 1. (a) The hydrogen atom wave functions are written as Unim. State the values of n, I, and m. State the relation between a physical quantity and each quantum number. At =0 the hydrogen atom is in the superposition state (7,0) = 4200 + A¥210 V3 where A is a real positive constant. Find A by normalization and determine the wave function at time t > 0. Find the average energy of the electron in eV given...
1-3i An electron is in the spin state x A a. Determine the normalization constant A b. Find the expectation value for S c. Find the expectation values for Sx d. Find the expectation values for Sz e. If you measure S on this electron, what values you get, and what is the probability of cachn? f. If you measure on this electron, what values you get, and what is the probability of cachn? g. If you measure Sy on...
- A() An electron is in the spin state x= A Determine the normalisation constant 1 i A, and then find the expectation value of the z-component of the spin operator h [6]
4. Spin (10 marks) Suppose an electron is in a state such that its spin can be described by a linear superposition of the eigenspinors of S +A 32 2/22 (a) Normalise the state. (b) What are the possible outcomes of a measurement of the z-component of the spin? What is the probability of each possible result? (c) What are the expectation value and uncertainty of the z-component of the spin? (d) What are the possible results of measuring the...
3 of 5 Problem #2 (show all your work!) spin For the state +), calculate the expectation values and uncertainties for measurements of S, Sy, and S,
4. A (one dimensional) particle in a box of length 2a (i.e., zero potential energy) is represented by the wavefunction v(x) 0, otherwise a. Sketch the wavefunction. Write down the (time independent) Schrodinger equation. Show whether or not the wavefunction is a solution to the equation. b. What does it mean physically if the wavefunction of the particle is NOT a solution to the Schrodinger equation? Explain. c. Determine the normalization constant A. 5. Same system. Find the average or...
3- A one-dimensional harmonic oscillator wave function is ψ(x) = Axe-bx2 a) Find the total energy E b) Find the constant b c) Find the normalization constant A. d) Find the expectation value of x, e) Find the uncertainty in x, Ох. f) Find the expectation value of p g) Find the uncertainty in p, Op For the Hamiltonian matrix shown below:
3- A one-dimensional harmonic oscillator wave function is ψ(x) = Axe-bx2 a) Find the total energy E b)...
b 2. Suppose a spin-2 particle is in the state that particle. 2,0) + 2,1) Find the expectation value of S, for D 3. In the t spin-1/2 basis, consider the two operators 2 1 12d B- (2 i A= ni 2 (a) Find the commutator [A, B (b) Suppose we measure a number of particles in state |t), using A and B. Find the average values (A) and (B) from these measurements. (c) Use the uncertainty principle to find...
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...
lsa(1) lsB(1) 1Isa(2) 1sja 7. Consider this two-electron wave function: ψ-C Write the expression for ψ that comes from expanding the determinant. Find the normalization constant, C. The 1s orbitals are orthonormal, and so are the spin orbitals. a) b) Using your answer from (a), show that the wave function factors into a spin part and a spatial part. Hint: It may help to rewrite each spin orbit so its spatial and spin factors are clearer. For instance, rewrite Isa...
ANSWER ALL QUESTIONS 1. (a) The hydrogen atom wave functions are written as Unim. State the values of n, I, and m. State the relation between a physical quantity and each quantum number. At =0 the hydrogen atom is in the superposition state (7,0) = 4200 + A¥210 V3 where A is a real positive constant. Find A by normalization and determine the wave function at time t > 0. Find the average energy of the electron in eV given...
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