Title: Expectation value of energy from the position state of hydrogen atom
Tags: quantum-mechanics;homework-and-exercises;wavefunction;atoms;hydrogen
Question: I was given with the position state of hydrogen atom:
R_{21} =\left(\sqrt{\frac{1}{3}}Y^0_1 + \sqrt{\frac{2}{3}}Y^1_1\right)
I am getting confused about getting the expectation value of energy, I know from the form that:
$$ R_{21} \Rightarrow n=2, l=1$$
according to this website.
I should just:
$$ \langle E_n\rangle = - \frac{1}{2} \alpha^2\mu c^2 \frac{\frac{1}{3} \cdot \frac{1}{2^2} + \frac{2}{3} \cdot \frac{1}{2^2}}{2^2}
\\= - \frac{1}{2} \alpha^2\mu c^2 \frac{\frac{1}{12} + \frac{2}{12}}{4}
\\= - \frac{1}{2} \alpha^2\mu c^2 \frac{\frac{3}{12}}{4}
\\= - \frac{1}{2} \alpha^2\mu c^2\frac{1}{16}$$
Am I right?
Comment: @BMS Please feel free to look at the same notation shown @ page 5 of this PDF: http://www.physics.udel.edu/~msafrono/626/Lecture%201.pdf

Comment: @BMS ok, sorry, I just contacted my prof, she said it was typo. sorry. I just switched it now. Can somebody confirm if my answer is correct?

Comment: I think he means superposition.  "Y" usually represents the angular (spherical harmonic)contribution to the wavefunction, with the subscript being "l" and superscript being "m". Then it is common to refer to linear combinations of different Ys.

Comment: Where were you given that position state? It looks a bit odd. Homework? A book?


Here is another answer: $R_{12}$ ==>n=1,l=2 (now corrected)
In the hydrogen atom, if $n$ = 1 then $l$ = 0
$0 \leq l \leq n-1$
$\psi = RY$
R is the radial function. Y is the spherical harmonic function. You can not say R = Y. You can not say R is a linear combination of Ys.
Also, in the n=2 state, the energy should be $\frac{1}{4}$ that of the ground state. The division by $2^2$ looks like a mistake.
Comment for this answer: Yes, I am really really confused with this notation, also, please look at page 5 of this pdf, which shows the use of such notation: http://www.physics.udel.edu/~msafrono/626/Lecture%201.pdf
Comment for this answer: ok, sorry, I just contacted my prof, she said it was typo. sorry. I just switched it now. Can somebody confirm if my answer is correct?
Comment for this answer: So sir, we cannot get the expectation value of energy given the radial wavefunction shown?
Comment for this answer: that correct is definitely progress, but what you have written in the question still doesn't make sense. $\psi = RY$ . R is the radial function. Y is the spherical harmonic function. You can not say R = Y. You can not say R is a linear combination of Ys.
Comment for this answer: it seem there are still two seperate mistakes. 1. The way you describe the problem still doesn't make sense. You are basically stating that the radial function equals the spherical harmonic function, which makes no sense. 2. I can't be sure because the problem still doesn't make sense, but it seems you are off by a factor of 4.

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