Spin Hamiltonian of a hydrogen atom in a magnetic field

In summary: However, if you are not familiar with this notation, it can be difficult to understand what is happening. For example, if you try to evaluate the operator ##\hat S_{1z }## on the state ##|\omega_0 \hat S_{1z }|1\rangle##, you may not be able to do so because you are not sure what the state ##|\omega_0 \hat S_{1z }|1\rangle## represents. In summary, one difficulty people have when working with the operator ##\hat S_{1z }## is understanding the notation for the state ##|1\rangle##.
  • #1
Nullity
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Homework Statement
5.1 of the first picture is the homework statement and the second is the solution. So my question is in the solution, how did they compute <1|w_0S_z|1>?
Relevant Equations
The book uses Dirac notation for matrix computation but this homework is not for a class and is strictly for recreation so feel free to solve it using the mathematical methods you feel comfortable with.
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qa
 
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  • #2
Can you give us some indication of why you are having trouble evaluating ##\langle 1 |\omega_0 \hat S_{1z }|1\rangle##? If you have made an attempt, please show us what you have done so we can tell where you are having difficulty. If you cannot get started with an attempt, then it is likely that you are having trouble with the notation. For example, are you clear on the meaning of the state ##|1 \rangle##? Can you describe the spins of the electron and the proton in this state?

One possible source of confusion is with the notation of the operator ##\hat S_{1z }## . The subscript 1 denotes the electron. But the ##1## in the state ##|1\rangle## denotes a particular state of the electron-proton system. When ##\hat S_{1z }## acts on the state ##|1 \rangle##, the operator ##\hat S_{1z }## acts only on the electron part of the state ##|1\rangle##. It is understood that this operator does not act on the proton. Or, in other words, the operator ##\hat S_{1z }## acts as the identity operator ##\mathbb { I}_2## when acting on the proton part of the state. The subscript 2 here refers to the proton. The proper way to write the second term of the Hamiltonian would be as a direct product of operators: ##\hat S_{1z } \otimes \mathbb {I}_2 ##. But for convenience, the ##\otimes \mathbb {I}_2## part is not written and is assumed to be understood.

Thus, consider a state ##|\Psi \rangle = |\alpha_1 \rangle |\beta_2 \rangle## in which the electron is in some spin state ##|\alpha_1 \rangle## while the proton is in spin state ##|\beta_2 \rangle##. Then $$\langle \Psi|\hat S_{1z}|\Psi \rangle = \langle \beta_2|\langle \alpha_1|\hat S_{1z }\otimes \mathbb {I}_2|\alpha_1 \rangle |\beta_2 \rangle = \langle \alpha_1|\hat S_{1z }|\alpha_1\rangle \langle \beta_2|\mathbb {I}_2|\beta_2\rangle = \langle \alpha_1|\hat S_{1z }|\alpha_1\rangle \langle \beta_2|\beta_2\rangle$$When you get used to this, you will mentally go directly from the far left to the far right of this series of steps without having to write the intermediate steps.
 

1. What is a spin Hamiltonian?

A spin Hamiltonian is a mathematical representation of the energy levels and interactions of the spin of an atom or molecule in a magnetic field. It is used in quantum mechanics to describe the behavior of particles with spin, such as electrons and protons.

2. How does a magnetic field affect the spin of a hydrogen atom?

A magnetic field causes the spin of a hydrogen atom to split into two energy levels, called the spin-up and spin-down states. This is known as the Zeeman effect and is represented by the spin Hamiltonian.

3. What is the significance of the spin Hamiltonian in studying hydrogen atoms?

The spin Hamiltonian is important in understanding the behavior of hydrogen atoms in a magnetic field. It allows scientists to predict and analyze the energy levels, transitions, and magnetic properties of hydrogen atoms, which are essential in many areas of physics and chemistry.

4. How is the spin Hamiltonian of a hydrogen atom calculated?

The spin Hamiltonian of a hydrogen atom is calculated using the Pauli matrices, which represent the possible spin states of the atom. The Hamiltonian is then solved using quantum mechanics equations to determine the energy levels and interactions of the spin in a magnetic field.

5. Can the spin Hamiltonian be applied to other atoms or molecules?

Yes, the spin Hamiltonian can be applied to any atom or molecule with spin in a magnetic field. However, the specific equations and calculations may vary depending on the properties of the atom or molecule being studied.

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