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Better explanation of ket notation

  • #1

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 [tex]ψ> = (1/√2) 1> - (1/2) 0> + (1/2) -1> [/tex] Find Ly.

b. A spin 1/2 is in the state [tex]ψ> = (1+i)/3 +> + (1/√3) -> [/tex] Calculate <Sz> and <Sx>, find the probabilities of finding [tex]± \hbar/2[/tex] 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!
 

Answers and Replies

  • #2
649
3

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 [tex]ψ> = (1/√2) 1> - (1/2) 0> + (1/2) -1> [/tex] Find Ly.

b. A spin 1/2 is in the state [tex]ψ> = (1+i)/3 +> + (1/√3) -> [/tex] Calculate <Sz> and <Sx>, find the probabilities of finding [tex]± \hbar/2[/tex] 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 [itex]\langle L_y \rangle [/itex]. Remember that [itex] \langle L_y \rangle [/itex] means exactly what it looks like [itex] \langle \psi |L_y |\psi \rangle [/itex]. So you are interested in sandwiching the operater [itex]L_y [/itex] in between your state. So the next step is to find a useful representation for [itex]L_y[/itex]. Since you state is given in terms of quantum number [itex]l[/itex], perhaps look for expressions of [itex]L_y[/itex] that can manipulate those types of states.
 
  • #3
When you say find Ly, I imagine you mean [itex]\langle L_y \rangle [/itex]. Remember that [itex] \langle L_y \rangle [/itex] means exactly what it looks like [itex] \langle \psi |L_y |\psi \rangle [/itex]. So you are interested in sandwiching the operater [itex]L_y [/itex] in between your state. So the next step is to find a useful representation for [itex]L_y[/itex]. Since you state is given in terms of quantum number [itex]l[/itex], perhaps look for expressions of [itex]L_y[/itex] that can manipulate those types of states.
Unless it's a typo (very possible with this prof), the problem asks for [itex]L_y [/itex]. I looked through griffith's ch.4, and the closest I could come up with is [itex] L^2*ψ = \hbar^2*l(l+1)ψ[/itex], which worries me, as that would introduce[itex]L_x^2 [/itex] and [itex]L_z^2 [/itex]. My problem with the given information stands though, even if I assemble [itex] <(1/√2) 1|L_y|(1/√2) 1> - <(1/2) 0|L_y|(1/2) 0> + <(1/2) -1|L_y|(1/2) -1> [/itex], I don't know what those numbers mean, and I haven't found a source that explains them.
 
  • #4
vela
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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 [tex]ψ> = (1/√2) 1> - (1/2) 0> + (1/2) -1> [/tex] 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 [tex]ψ> = (1+i)/3 +> + (1/√3) -> [/tex] Calculate <Sz> and <Sx>, find the probabilities of finding [tex]± \hbar/2[/tex] 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.)
 

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