The Hamiltonian of hydrogen atom is given by H = -(h²/2m)V² - e²/2. Find the first and second energy values using trial function = Ae¯ar.
The Hamiltonian of hydrogen atom is given by H = -(h²/2m)V² - e²/2. Find the first and second energy values using trial function = Ae¯ar.
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![### Problem Statement: Hamiltonian of Hydrogen Atom
The Hamiltonian of a hydrogen atom is given by the equation:
\[ H = -\left(\frac{\hbar^2}{2m}\right)\nabla^2 - \frac{e^2}{2} \]
where:
- \( \hbar \) is the reduced Planck's constant,
- \( m \) is the mass of the electron,
- \( \nabla^2 \) is the Laplacian operator,
- \( e \) is the elementary charge.
To find the first and second energy values, use the trial wave function:
\[ \psi = Ae^{-\alpha r} \]
where:
- \( A \) is the normalization constant,
- \( \alpha \) is a parameter,
- \( r \) is the radial distance from the nucleus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F18d7602f-9f25-445c-bda8-4d38e99e2d82%2F62dcac46-a10f-4ad6-b1f4-ae557406006e%2Fcxsycht_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement: Hamiltonian of Hydrogen Atom
The Hamiltonian of a hydrogen atom is given by the equation:
\[ H = -\left(\frac{\hbar^2}{2m}\right)\nabla^2 - \frac{e^2}{2} \]
where:
- \( \hbar \) is the reduced Planck's constant,
- \( m \) is the mass of the electron,
- \( \nabla^2 \) is the Laplacian operator,
- \( e \) is the elementary charge.
To find the first and second energy values, use the trial wave function:
\[ \psi = Ae^{-\alpha r} \]
where:
- \( A \) is the normalization constant,
- \( \alpha \) is a parameter,
- \( r \) is the radial distance from the nucleus.
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