A strange wave function of the Hydrogen atom

In summary, the conversation discusses the possible values of the measurement of L_z, the probability of a measurement of L^2 giving the value 6 \hbar^2, and the minimum value of an energy measurement for an electron in a H atom. The only possible measurement for L_z is \hbar, but there is uncertainty about the reasoning behind this. To solve the problem, the wavefunction can be rewritten as a linear combination of stationary states, but the square on the r part and the exponentiation of the angular part pose difficulties. To find the coefficient of a particular eigenfunction, one does not need to completely decompose the wavefunction. However, it is unclear how to find a definite number with the n dependence.
  • #1
omegax241
2
2
I am trying to solve the following exercise.

In a H atom the electron is in the state described by the wave function in spherical coordinates:

[tex] \psi (r, \theta, \phi) = e^{i \phi}e^{-(r/a)^2(1- \mu\ cos^2\ \theta)} [/tex]

With [itex]a[/itex] and [itex]\mu[/itex] positive real parameters. Tell what are the possible values of the measurement of [itex]L_z[/itex], what is the probability that a measurement of [itex]L^2[/itex] gives the value [itex]6 \hbar^2[/itex], and then what is the minimum value of an energy measurement.

First I've observed that [itex]m=1[/itex], so the only possible measurement for [itex]L_z[/itex] is [itex]\hbar[/itex]. (I'm pretty confident with this reasoning, but not so much.)For the other questions my idea was to rewrite the wavefunction as a linear combination of stationary states of the form [itex]\phi_{n,l,1}[/itex], but the problem is the square on the [itex]r[/itex] part, and the fact that the angular part [itex]\theta[/itex] is exponentiated.
In fact I've even started to ask myself is if necessary at all to find this decomposition to solve the problem, but nothing comes to mind.
Thank you for listening, every bit of help is appreciated.
 
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  • #2
omegax241 said:
I am trying to solve the following exercise.

In a H atom the electron is in the state described by the wave function in spherical coordinates:

[tex] \psi (r, \theta, \phi) = e^{i \phi}e^{-(r/a)^2(1- \mu\ cos^2\ \theta)} [/tex]

With [itex]a[/itex] and [itex]\mu[/itex] positive real parameters. Tell what are the possible values of the measurement of [itex]L_z[/itex], what is the probability that a measurement of [itex]L^2[/itex] gives the value [itex]6 \hbar^2[/itex], and then what is the minimum value of an energy measurement.

First I've observed that [itex]m=1[/itex], so the only possible measurement for [itex]L_z[/itex] is [itex]\hbar[/itex]. (I'm pretty confident with this reasoning, but not so much.)For the other questions my idea was to rewrite the wavefunction as a linear combination of stationary states of the form [itex]\phi_{n,l,1}[/itex], but the problem is the square on the [itex]r[/itex] part, and the fact that the angular part [itex]\theta[/itex] is exponentiated.
In fact I've even started to ask myself is if necessary at all to find this decomposition to solve the problem, but nothing comes to mind.
Thank you for listening, every bit of help is appreciated.
To find the coefficient of a particular eigenfunction you do not need to completely decompose the wavefunction. Do you know how to do that?
 
  • #3
omegax241 said:
Thank you for listening, every bit of help is appreciated.
I've had a look at this. I must admit I don't see how to tackle it. That wave function looks horrible. Sorry, I've no ideas either.
 
  • #4
PeroK said:
To find the coefficient of a particular eigenfunction you do not need to completely decompose the wavefunction. Do you know how to do that?

Yes, I should evaluate the product:
[tex] \langle \phi_{n, l ,m} | \psi \rangle [/tex]
But suppose I want to tackle the second question, if a value of [itex]6 \hbar[/itex] is found for [itex]L^2[/itex] this means that [itex]l = 2[/itex], and then [itex] m = \pm 1 ; \pm 2 ; 0[/itex]. So I should evaluate the coefficents​
[tex] \langle \phi_{n, 2, \pm 1} | \psi \rangle [/tex]
[tex] \langle \phi_{n, 2, \pm 2} | \psi \rangle [/tex]
[tex] \langle \phi_{n, 2, 0} | \psi \rangle [/tex]

But how can I find a definite number with this n dependence ?
 
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  • #5
omegax241 said:
Yes, I should evaluate the product:
[tex] \langle \phi_{n, l ,m} | \psi \rangle [/tex]
But suppose I want to tackle the second question, if a value of [itex]6 \hbar[/itex] is found for [itex]L^2[/itex] this means that [itex]l = 2[/itex], and then [itex] m = \pm 1 ; \pm 2 ; 0[/itex]. So I should evaluate the coefficents​
[tex] \langle \phi_{n, 2, \pm 1} | \psi \rangle [/tex]
[tex] \langle \phi_{n, 2, \pm 2} | \psi \rangle [/tex]
[tex] \langle \phi_{n, 2, 0} | \psi \rangle [/tex]

But how can I find a definite number with this n dependence ?
You know that ##m = 1##. It's the variable ##n## that's the problem.
 
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  • #6
Doesn't parity tell you that an eigenstate of l=2 must be even? The wave function provided is odd.
 
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Related to A strange wave function of the Hydrogen atom

1. What is a wave function?

A wave function is a mathematical representation of the quantum state of a particle, which describes the probability of finding the particle in a particular location or state.

2. Why is the wave function of the Hydrogen atom considered strange?

The wave function of the Hydrogen atom is considered strange because it does not have a definite shape or size like classical objects. Instead, it exists in a superposition of multiple states, making it difficult to visualize or understand in traditional terms.

3. How is the wave function of the Hydrogen atom used in quantum mechanics?

The wave function of the Hydrogen atom is used in quantum mechanics to calculate the probability of finding an electron in a particular energy level or orbital. It also helps to predict the behavior of the atom and its interactions with other particles.

4. Can the wave function of the Hydrogen atom be observed or measured?

No, the wave function of the Hydrogen atom cannot be directly observed or measured. It is a mathematical concept that represents the quantum state of the atom, and its values can only be obtained through calculations and experiments.

5. How does the wave function of the Hydrogen atom relate to the uncertainty principle?

The wave function of the Hydrogen atom is directly related to the uncertainty principle, which states that it is impossible to know both the exact position and momentum of a particle simultaneously. The wave function represents the probability of finding the particle in a particular location, and the uncertainty principle explains why the position of an electron cannot be precisely determined.

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