Expectation value of 1s state of hydrogen driving me absolutely nuts

The correct evaluation of the expectation value for the 1s state of hydrogen is 3/2 times the Bohr radius, which disagrees with the result found using the Bohr theory. This discrepancy may be due to the fact that <1/r> and <r> are two different mathematical notations for the same quantity. In summary, the expectation value <1/r> for the 1s state of hydrogen is 3/2 times the Bohr radius, which differs from the result obtained using the Bohr theory. This could be attributed to the use of different mathematical notations for the same quantity.
  • #1
doubleB
19
0
My question says:

"Evaluate the expectation value <1/r> of the 1s state of hydrogen. How does this result compare to the result found using the Bohr theory?"

Firstly, I have been told that <1/r> does not mean <1/r> but rather that it means 1/<r>. Having made it this far I now do the 1/<r> and get about 12 times the bohr radius which is obviously wrong. Apparently the answer is 3/2 the bohr radius.
But also, I thought that doing <r> was supposed to give you 3/2 bohr radius, so what is this 1/<r> for?
Basically, what the hell is going on, in simple, clear terms?
Thank you for any help,
Ben
 
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  • #2
1/<r> is wrong. It should be <1/r>.
 

Related to Expectation value of 1s state of hydrogen driving me absolutely nuts

1. What is the expectation value of the 1s state of hydrogen?

The expectation value of the 1s state of hydrogen is the average value that would be obtained if the measurement of the energy of the electron in the 1s orbital was repeated many times. It is a measure of the most probable energy of the electron in this state.

2. How is the expectation value of the 1s state of hydrogen calculated?

The expectation value of the 1s state of hydrogen is calculated by taking the integral of the probability density function (wavefunction squared) multiplied by the energy operator over all possible values of energy.

3. What does the expectation value of the 1s state of hydrogen tell us about the electron's energy?

The expectation value of the 1s state of hydrogen tells us the most probable energy of the electron in this state. This means that if we were to measure the energy of the electron in the 1s orbital, we would most likely obtain the expectation value as the result.

4. How does the expectation value of the 1s state of hydrogen change with different quantum numbers?

The expectation value of the 1s state of hydrogen is dependent on the principal quantum number, n. As n increases, the expectation value also increases, indicating that the electron is further away from the nucleus and therefore has a higher average energy.

5. Why is the expectation value of the 1s state of hydrogen important in quantum mechanics?

The expectation value of the 1s state of hydrogen is important in quantum mechanics because it allows us to predict the most probable energy of the electron in this state. This is crucial in understanding the behavior and properties of atoms and molecules, and is a fundamental concept in quantum mechanics.

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