Two particle state
Two Particle State problem, measurement of total angular momentum, finding probabilities, action of lowering operator
a) A particle of spin \(3 / 2\) and a particle of spin 1 are found in the state \(\left|\frac{3}{2}-\frac{1}{2}\right\rangle\) of total spin. Assume that the particles are in a state of zero orbital angular momentum.
Find which values of the z-component of the spin of the two particles can be measured and with what probabilities.
The following information applies to parts \(\mathbf{b}\) ) and \(\mathbf{c}\) ).
An electron in the hydrogen atom occupies the following position and spin state:
$$ \Psi=\frac{1}{5}\left(3 \psi_{322} \chi_{-}+4 \psi_{321} \chi_{+}\right) $$
It can be also written as:
$$ \begin{aligned} &\Psi=R_{32} \frac{1}{5}\left(3 Y_{2}^{2} \chi_{-}+4 Y_{2}^{1} \chi_{+}\right) \\ &\left.\left.|\Psi\rangle=R_{32} \frac{1}{5}(3(122\rangle \otimes|1 / 2-1 / 2\rangle)+4(121\rangle \otimes|1 / 21 / 2\rangle\right)\right) \end{aligned} $$
b) If you measure the total angular momentum, what values would you get and with what probabilities?
c) Find the action of the lowering operator \(J_{-}=L_{-}+S_{-}\)on the wave function.
Homework Answers
A particle initially in the state has the position-space wavefunction
A particle initially in the state \(|\psi\rangle\) has the position-space wavefunction$$ \psi(x)=N e^{-x^{2} / 8 a^{2}} $$(a) What is the normalization coefficient \(N\) ?(b) The state is then altered. Find the position-space wavefunction of the altered state$$ \left|\psi^{\prime}\right\rangle=e^{-i p c / A}|\psi\rangle $$(c) Calculate the expectation values of \((\ell)\), for both \(|\psi\rangle\) and \(\left|\psi^{\prime}\right\rangle\).
2. Addition of Angular Momentum a) (8pts) Given two spin 1/2 particles, what are the four possibilities for their spin configuration? Put your answer in terms of states such as | 11). where the f...
2. Addition of Angular Momentum a) (8pts) Given two spin 1/2 particles, what are the four possibilities for their spin configuration? Put your answer in terms of states such as | 11). where the first arrow denotes the z-component of the particle's spin. Identify the m values for each state. b)(7pts) If you apply the lowering operator to a state you get Apply the two-state lowering operator S--S(,) +S(), where sti) acts on the first state and S acts on...
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System A consists of two spin-1/2 particles, and has a four-dimensional Hilbert space. 1. Write down...
System A consists of two spin-1/2 particles, and has a four-dimensional Hilbert space. 1. Write down a basis for the Hilbert space of two spin-1/2 particles. 2. Calculate the matrix of the angular momentum operator, Sfot = (ŜA, ŠA, ŜA) for system A, in the basis of question 4A.1, and express them in this basis. 3. Calculate the square of the total angular momentum of system A , Spotl?, and express this operator in the basis of question 4A.1. 4....
I have the first method complete, but I can't figure out thesecond method. I have...
I have the first method complete, but I can't figure out the
second method Could
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3. (6 points) Measurements on a two-particle state Consider the state for a system of two...
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b 2. Suppose a spin-2 particle is in the state that particle. 2,0) + 2,1) Find...
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A particle initially in the state \(|\psi\rangle\) has the position-space wavefunction$$ \psi(x)=N e^{-x^{2} / 8 a^{2}} $$(a) What is the normalization coefficient \(N\) ?(b) The state is then altered. Find the position-space wavefunction of the altered state$$ \left|\psi^{\prime}\right\rangle=e^{-i p c / A}|\psi\rangle $$(c) Calculate the expectation values of \((\ell)\), for both \(|\psi\rangle\) and \(\left|\psi^{\prime}\right\rangle\).
2. Addition of Angular Momentum a) (8pts) Given two spin 1/2 particles, what are the four possibilities for their spin configuration? Put your answer in terms of states such as | 11). where the first arrow denotes the z-component of the particle's spin. Identify the m values for each state. b)(7pts) If you apply the lowering operator to a state you get Apply the two-state lowering operator S--S(,) +S(), where sti) acts on the first state and S acts on...
System A consists of two spin-1/2 particles, and has a four-dimensional Hilbert space. 1. Write down a basis for the Hilbert space of two spin-1/2 particles. 2. Calculate the matrix of the angular momentum operator, Sfot = (ŜA, ŠA, ŜA) for system A, in the basis of question 4A.1, and express them in this basis. 3. Calculate the square of the total angular momentum of system A , Spotl?, and express this operator in the basis of question 4A.1. 4....
I have the first method complete, but I can't figure out the
second method Could
someone please show how to use the second method?2. Find the unit step response of:$$ \begin{aligned} \dot{\overrightarrow{\mathbf{x}}}(t) &=\left[\begin{array}{cc} 0 & 1 \\ -2 & -2 \end{array}\right] \overrightarrow{\mathbf{x}}(t)+\left[\begin{array}{l} 1 \\ 1 \end{array}\right] u(t) \\ y(t) &=\left[\begin{array}{cc} 2 & 3 \end{array}\right] \overrightarrow{\mathbf{x}}(t) \end{aligned} $$by two methods (1): transfer function and then (2) \(y(t)=\mathbf{C} e^{\mathbf{A} t} \overrightarrow{\mathbf{x}}(0)+\mathbf{C} \int_{0}^{t} e^{\mathbf{A}(t-\tau)} \mathbf{B} u(\tau) d \tau+\mathbf{D} u(t)\). Re-member that the Laplace...
3. (6 points) Measurements on a two-particle state Consider the state for a system of two spin-1/2 particles, (2]+).I+)2 +1-)[+)2-1-)1-)2). (a) Show that this state is normalized. (b) What is the probability of measuring S: (the z-component of spin for particle 1) to be +h/2? After this measurement is made with this result, what is the state of the system? If we make a measurement in this new state, what is now the probability of measuring S3 = +h/2? (e)...
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...
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