Is this state description normalized?

[bowlbase]

Homework Statement

These are rather simple questions but the rules for all of this are not quite clear to me yet. I'm to determine whether or not the following states are "legal" and if not I should normalize them.

a. $\frac{1}{√385} ∑_{x=1}^{10}x^2 |x>$

b. $\frac{1}{√2} |u_x>+\frac{1}{√2} |u_z>$

c.$e^{0.32i}(0.01|0>+0.25|1>+0.16|2>$

Homework Equations

The Attempt at a Solution

For a I believe I am correct in in simply squaring the fraction and the sum giving me a number much greater than 1 (≈385). So, in that case, the normalization factor 'N' would just be $\frac{1}{385}$ in order to make it 1.

For b my lack of understanding is obvious. At first glance I can see that the equation is normalized but the bases are not the same. In class we switch bases in this case but I don't know if I can say the equation is normalized as it is. Is this a legal description of the state that I can say is normalized or is it necessary to switch to common bases and introduce a normalization factor?

Lastly c, I believe is fine as it is. However, I just want to get confirmation on what the exponential actually is. So far in class we have ignored exponentials and the complex 'i'. Are these just some sort of phase constants?

 

[nrqed]

Homework Statement

These are rather simple questions but the rules for all of this are not quite clear to me yet. I'm to determine whether or not the following states are "legal" and if not I should normalize them.

a. $\frac{1}{√385} ∑_{x=1}^{10}x^2 |x>$

b. $\frac{1}{√2} |u_x>+\frac{1}{√2} |u_z>$

c.$e^{0.32i}(0.01|0>+0.25|1>+0.16|2>$

Homework Equations

The Attempt at a Solution

For a I believe I am correct in in simply squaring the fraction and the sum giving me a number much greater than 1 (≈385). So, in that case, the normalization factor 'N' would just be $\frac{1}{385}$ in order to make it 1.

What is the inner product between those kets? I assume it is $\langle i|j\rangle = \delta_{ij}$? We need that info to help you.
Is is exactly equal to 385? Did you take into account the factor $1/\sqrt{385}$ in front?

For b my lack of understanding is obvious. At first glance I can see that the equation is normalized but the bases are not the same. In class we switch bases in this case but I don't know if I can say the equation is normalized as it is. Is this a legal description of the state that I can say is normalized or is it necessary to switch to common bases and introduce a normalization factor?

We need the inner product between these kets.

Lastly c, I believe is fine as it is. However, I just want to get confirmation on what the exponential actually is. So far in class we have ignored exponentials and the complex 'i'. Are these just some sort of phase constants?

It is just a complex phase, as in complex analysis. EDIT: what do you get for the inner product?

 

[bowlbase]

I guess I just assumed these were common.

$\langle u_z|u_x\rangle$
$|u_x\rangle=\frac{1}{√2} |u_z\rangle + \frac{1}{√2} |d_z\rangle$
$\frac{1}{√2}(\langle u_z|u_z\rangle) + \frac{1}{√2}((\langle u_z|d_z\rangle)$
$\frac{1}{√2}(1) + \frac{1}{√2}(0)$

So $\langle u_z|u_x\rangle = \frac{1}{√2}$

For a the quantity of the sum times the sqrt = \frac{385}{√385}. Squared that is 385. So
$N(385)=1$

Sorry, for the last one I wasn't thinking. Going to look at it for a minute.

 

[nrqed]

I guess I just assumed these were common.

$\langle u_z|u_x\rangle$
$|u_x\rangle=\frac{1}{√2} |u_z\rangle + \frac{1}{√2} |d_z\rangle$
$\frac{1}{√2}(\langle u_z|u_z\rangle) + \frac{1}{√2}((\langle u_z|d_z\rangle)$
$\frac{1}{√2}(1) + \frac{1}{√2}(0)$

So $\langle u_z|u_x\rangle = \frac{1}{√2}$

Ah, they are spin components. ok. Then you have everything to calculate if it is normalized

For a the quantity of the sum times the sqrt = \frac{385}{√385}. Squared that is 385. So
$N(385)=1$

Sorry, for the last one I wasn't thinking. Going to look at it for a minute.

Ok, no problem!

 

[bowlbase]

So for the spin question, the bases do matter then? I just need to switch to a common base?

 

[nrqed]

So for the spin question, the bases do matter then? I just need to switch to a common base?

Yes, the basis matters because the U's along different axes are not orthonormal. You have two choices: you may re-express everything in terms of one coordinate or, much faster, calculate directly the norm of your state. Since you worked out the inner product between $| u_x \rangle$ and $| u_z \rangle$, the calculation is only two lines.

 

[bowlbase]

okay, thanks. I think I understand now. I appreciate the help.

 

[nrqed]

okay, thanks. I think I understand now. I appreciate the help.

You are welcome!