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The image presents a question related to quantum numbers derived from the Schrödinger equation for the hydrogen atom. Let’s break it down and answer the blanks.

### Quantum numbers:
1. **Principal quantum number (n)**: Determines the energy level of the electron.
2. **Azimuthal quantum number (l)**: Determines the shape of the orbital, where \( l \) ranges from 0 to \( n-1 \).
3. **Magnetic quantum number (m_l)**: Determines the orientation of the orbital, and \( m_l \) ranges from \( -l \) to \( +l \).

### For \( n = 1 \):
- The possible values for \( l \) range from **0 to 0** (since \( l \) ranges from 0 to \( n-1 \), and if \( n = 1 \), \( l \) can only be 0).
- The total number of orbitals at the \( n = 1 \) level is **1**, because the number of orbitals is given by the possible \( m_l \) values, and for \( l = 0 \), \( m_l \) can only be 0 (one orbital).

### For \( l = 0 \):
- The possible values for \( m_l \) range from **0 to 0** (since \( m_l \) ranges from \( -l \) to \( +l \), and \( l = 0 \) only allows \( m_l = 0 \)).
- The total number of orbitals at the \( l = 0 \) sublevel is **1**, because there is only one possible \( m_l \) value.

#### Completed answers:
- For \( n = 1 \): 
   - The quantum number \( l \) can have values from **0 to 0**.
   - The total number of orbitals at \( n = 1 \) is **1**.
   
- For \( l = 0 \): 
   - The quantum number \( m_l \) can have values from **0 to 0**.
   - The total number of orbitals at \( l = 0 \) is **1**.

Would you like more details, or do you have any other questions about quantum numbers?

### 5 Related Questions:
1. What are the possible values of \( l \) when \( n = 2 \)?
2. How do the quantum numbers relate to the shape of atomic orbitals?
3. What is the significance of the magnetic quantum number \( m_l \)?
4. How do quantum numbers determine the electron configuration of an atom?
5. What are the possible values for \( m_s \) (the spin quantum number)?

### Tip:
The total number of orbitals in a given energy level \( n \) is equal to \( n^2 \). For example, for \( n = 2 \), there are \( 2^2 = 4 \) orbitals.

If the value of n = 1, what are the possible values of the quantum number l, and how many orbitals exist at the n = 1 energy level? If the value of l = 0, what are the possible values of the magnetic quantum number ml, and how many orbitals are there at the l = 0 sublevel?

Explore the quantum numbers for the hydrogen atom's orbitals using Schrödinger's equation. Understand how the quantum number l can range from 0 to 0 when n = 1, and how this results in a single orbital at the n = 1 energy level. For l = 0, the magnetic quantum number ml can only be 0, yielding one orbital at the l = 0 sublevel.

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