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I'm reading Sakurai's Modern Quantum Mechanics. In page 99, there is a concept of Directional Eigenket. It doesn't have a definition, or any properties about it.

Because the angular dependence is common to all problems with spherical symmetry, we can isolate it and consider $$\langle \hat{n}|l,m\rangle=Y_l^m(\theta,\phi)=Y_l^m(\hat{n})$$ where we have defined a directional eigenket $|\hat{n}\rangle$. From this point of view, $Y_l^m(\theta, \phi)$ is the amplitude for the state characterized by $l,m$ to be found in the direction $\hat{n}$ specified by $\theta$ and $\phi$.

At first I thought it was the eigenket of the operator measuring the directional unit vector.(Probably it is similar or the same) But as I'm understanding it, (in position space) any function that only has a nonzero value in one direction will be such a eigenfunction. So I'm wondering,

Where is my understanding wrong, and What is this directional eigenket represented as in position space? Also, $|l,m\rangle$ refers to any eigenstate with $l,m$, right??(a linear combination of $|n,l,m\rangle$s) Degeneracy makes me a little confused here.

I'm reading Sakurai's Modern Quantum Mechanics. In page 99, there is a concept of Directional Eigenket. It doesn't have a definition, or any properties about it.

Because the angular dependence is common to all problems with spherical symmetry, we can isolate it and consider $$\langle \hat{n}|l,m\rangle=Y_l^m(\theta,\phi)=Y_l^m(\hat{n})$$ where we have defined a directional eigenket $|\hat{n}\rangle$. From this point of view, $Y_l^m(\theta, \phi)$ is the amplitude for the state characterized by $l,m$ to be found in the direction $\hat{n}$ specified by $\theta$ and $\phi$.

At first I thought it was the eigenket of the operator measuring the directional unit vector.(Probably it is similar or the same) But as I'm understanding it, (in position space) any function that only has a nonzero value in one direction will be such a eigenfunction. So I'm wondering,

Where is my understanding wrong, and What is this directional eigenket represented as in position space? Also, $|l,m\rangle$ refers to any eigenstate with $l,m$, right?  (a linear combination of $|n,l,m\rangle$s) Degeneracy makes me a little confused here.

I'm reading Sakurai's Modern Quantum Mechanics. In page 99, there is a concept of Directional Eigenket. It doesn't have a definition, or any properties about it.

Because the angular dependence is common to all problems with spherical symmetry, we can isolate it and consider $$\langle \hat{n}|l,m\rangle=Y_l^m(\theta,\phi)=Y_l^m(\hat{n})$$ where we have defined a directional eigenket $|\hat{n}\rangle$. From this point of view, $Y_l^m(\theta, \phi)$ is the amplitude for the state characterized by $l,m$ to be found in the direction $\hat{n}$ specified by $\theta$ and $\phi$.

At first I thought it was the eigenket of the operator measuring the directional unit vector.(Probably it is similar or the same) But as I'm understanding it, (in position space) any function that only has a nonzero value in one direction will be such a eigenfunction. So I'm wondering,

Where is my understanding wrong, and What is this directional eigenket represented as in position space? Also, $|l,m\rangle$ refers to any eigenstate with $l,m$, right?? Degeneracy makes me a little confused here.

I'm reading Sakurai's Modern Quantum Mechanics. In page 99, there is a concept of Directional Eigenket. It doesn't have a definition, or any properties about it.

Because the angular dependence is common to all problems with spherical symmetry, we can isolate it and consider $$\langle \hat{n}|l,m\rangle=Y_l^m(\theta,\phi)=Y_l^m(\hat{n})$$ where we have defined a directional eigenket $|\hat{n}\rangle$. From this point of view, $Y_l^m(\theta, \phi)$ is the amplitude for the state characterized by $l,m$ to be found in the direction $\hat{n}$ specified by $\theta$ and $\phi$.

At first I thought it was the eigenket of the operator measuring the directional unit vector.(Probably it is similar or the same) But as I'm understanding it, (in position space) any function that only has a nonzero value in one direction will be such a eigenfunction. So I'm wondering,

Where is my understanding wrong, and What is this directional eigenket represented as in position space? Also, $|l,m\rangle$ refers to any eigenstate with $l,m$, right??(a linear combination of $|n,l,m\rangle$s) Degeneracy makes me a little confused here.

I'm reading Sakurai's Modern Quantum Mechanics. In page 99, there is a concept of Directional Eigenket. It doesn't have a definition, or any properties about it. At first I thought it was the eigenket of the operator measuring the directional unit vector.(Probably it is similar or the same) But as I'm understanding it, (in position space) any function that only has a nonzero value in one direction will be such a eigenfunction. So I'm wondering,

Where is my understanding wrong, and What is this directional eigenket represented as in position space?

I'm reading Sakurai's Modern Quantum Mechanics. In page 99, there is a concept of Directional Eigenket. It doesn't have a definition, or any properties about it.

Because the angular dependence is common to all problems with spherical symmetry, we can isolate it and consider $$\langle \hat{n}|l,m\rangle=Y_l^m(\theta,\phi)=Y_l^m(\hat{n})$$ where we have defined a directional eigenket $|\hat{n}\rangle$. From this point of view, $Y_l^m(\theta, \phi)$ is the amplitude for the state characterized by $l,m$ to be found in the direction $\hat{n}$ specified by $\theta$ and $\phi$.

At first I thought it was the eigenket of the operator measuring the directional unit vector.(Probably it is similar or the same) But as I'm understanding it, (in position space) any function that only has a nonzero value in one direction will be such a eigenfunction. So I'm wondering,

Where is my understanding wrong, and What is this directional eigenket represented as in position space? Also, $|l,m\rangle$ refers to any eigenstate with $l,m$, right?? Degeneracy makes me a little confused here.