Better explanation of ket notation

[LarsPearson]

Homework Statement

I have two separate problems with the same issue; I don't grasp what information the equation of state is giving me.

a. A system with l=1 is in the state $$ψ> = (1/√2) 1> - (1/2) 0> + (1/2) -1>$$ Find Ly.

b. A spin 1/2 is in the state $$ψ> = (1+i)/3 +> + (1/√3) ->$$ Calculate <Sz> and <Sx>, find the probabilities of finding $$± \hbar/2$$ if spin is measured in z direction, and spin up if measured in x direction.

Homework Equations

The Attempt at a Solution

I can tell right away that the values are spin quantum numbers, and I assume both are superpositions of states, but I'm not sure what to DO with the given info to turn it into something usable. I have the griffiths textbook, but I'm not getting anything out of it or in my class notes about what the integers here represent, all my info is for vectors, and scouring the internet has so far failed me. If someone could point me to some relevant material/examples, or explain how to translate this(into matrices, I believe?) I'd be very grateful. Thanks!

 

[jfy4]

Homework Statement

I have two separate problems with the same issue; I don't grasp what information the equation of state is giving me.

a. A system with l=1 is in the state $$ψ> = (1/√2) 1> - (1/2) 0> + (1/2) -1>$$ Find Ly.

b. A spin 1/2 is in the state $$ψ> = (1+i)/3 +> + (1/√3) ->$$ Calculate <Sz> and <Sx>, find the probabilities of finding $$± \hbar/2$$ if spin is measured in z direction, and spin up if measured in x direction.

Homework Equations

The Attempt at a Solution

I can tell right away that the values are spin quantum numbers, and I assume both are superpositions of states, but I'm not sure what to DO with the given info to turn it into something usable. I have the griffiths textbook, but I'm not getting anything out of it or in my class notes about what the integers here represent, all my info is for vectors, and scouring the internet has so far failed me. If someone could point me to some relevant material/examples, or explain how to translate this(into matrices, I believe?) I'd be very grateful. Thanks!

When you say find Ly, I imagine you mean $\langle L_y \rangle$. Remember that $\langle L_y \rangle$ means exactly what it looks like $\langle \psi |L_y |\psi \rangle$. So you are interested in sandwiching the operater $L_y$ in between your state. So the next step is to find a useful representation for $L_y$. Since you state is given in terms of quantum number $l$, perhaps look for expressions of $L_y$ that can manipulate those types of states.

 

[LarsPearson]

When you say find Ly, I imagine you mean $\langle L_y \rangle$. Remember that $\langle L_y \rangle$ means exactly what it looks like $\langle \psi |L_y |\psi \rangle$. So you are interested in sandwiching the operater $L_y$ in between your state. So the next step is to find a useful representation for $L_y$. Since you state is given in terms of quantum number $l$, perhaps look for expressions of $L_y$ that can manipulate those types of states.

Unless it's a typo (very possible with this prof), the problem asks for $L_y$. I looked through griffith's ch.4, and the closest I could come up with is $L^2*ψ = \hbar^2*l(l+1)ψ$, which worries me, as that would introduce$L_x^2$ and $L_z^2$. My problem with the given information stands though, even if I assemble $<(1/√2) 1|L_y|(1/√2) 1> - <(1/2) 0|L_y|(1/2) 0> + <(1/2) -1|L_y|(1/2) -1>$, I don't know what those numbers mean, and I haven't found a source that explains them.

 

[vela]

Homework Statement

I have two separate problems with the same issue; I don't grasp what information the equation of state is giving me.

a. A system with l=1 is in the state $$ψ> = (1/√2) 1> - (1/2) 0> + (1/2) -1>$$ Find Ly.

The state is supposed to be written $\lvert \psi \rangle = \frac{1}{\sqrt{2}}\lvert 1 \rangle - \frac{1}{2}\lvert 0 \rangle + \frac{1}{2}\lvert -1 \rangle$.

b. A spin 1/2 is in the state $$ψ> = (1+i)/3 +> + (1/√3) ->$$ Calculate <Sz> and <Sx>, find the probabilities of finding $$± \hbar/2$$ if spin is measured in z direction, and spin up if measured in x direction.

Similarly, here you should have $\lvert\psi\rangle = \frac{1+i}{\sqrt{3}}\lvert + \rangle + \frac{1}{\sqrt{3}}\lvert - \rangle$.

Does that clear up your confusion about the numbers? (I'm not sure which numbers you're actually referring to in your last post.)

 

[paranormal]

The notation used in the given equations is called ket notation, which is commonly used in quantum mechanics to represent states of a system. In this notation, the symbol | > represents a state vector, and the numbers inside the brackets represent the quantum numbers associated with that state. In the first problem, the system has a quantum number of l=1, which corresponds to the angular momentum of the system. The state vector is a superposition of three different states, represented by the numbers 1, 0, and -1. The coefficients in front of each state represent the probability amplitudes of finding the system in that particular state.

To find Ly, you can use the ladder operators, which are mathematical operators that raise or lower the angular momentum quantum number by one. In this case, Ly is the ladder operator for the y-component of angular momentum. By using this operator on the given state vector, you can determine the y-component of the angular momentum.

In the second problem, the system is a spin 1/2, which means it has two possible states: spin up and spin down. The state vector is a superposition of these two states, with the coefficients representing the probability amplitudes of finding the system in each state. To find <Sz> and <Sx>, you can use the Pauli spin matrices, which are mathematical operators that represent the spin of a system in different directions. By using these operators on the state vector, you can determine the expected values of the spin in the z and x directions.

To find the probabilities of finding a specific spin state, you can use the Born rule, which states that the probability of finding a system in a particular state is equal to the squared magnitude of the coefficient in front of that state in the state vector. For example, to find the probability of finding spin up in the z direction, you would square the coefficient in front of the spin up state (1+i)/3.

I would suggest reviewing the mathematical concepts of quantum mechanics, such as operators, eigenvalues, and expectation values, to better understand how to work with ket notation. You can also consult other textbooks or online resources for further examples and explanations.