Product state and orthorgonality

In summary, the two states |0, e\rangle and |1\rangle|g\rangle are orthogonal because the probability of measuring one state while the system is in the other state is 0. This is due to the fact that the two states are not compatible and cannot coexist simultaneously.
  • #1
KFC
488
4
For two level atom trapped in a box (or cavity), initially at excited state without any photon inside, all possible states are

[tex]|0, e\rangle =|0\rangle|e\rangle \qquad and \qquad |1, g\rangle = |1\rangle|g\rangle[/tex]

e stands for excitated state, g stands for ground state.

Obviously,

[tex]\langle 0, e|0, e\rangle = 1[/tex]

and

[tex]\langle 1, g|1, g\rangle = 1[/tex]

I wonder if these two states [tex]|0, e\rangle[/tex] and [tex]|1\rangle|g\rangle[/tex] are orthorgonal? Why?

By the way, if I know the density operator at time T be [tex]\rho(t)[/tex], how to interpret [tex]\langle 0, e|\rho(t)|1, g\rangle[/tex] and [tex]\langle 1, g|\rho(t)|0, e\rangle[/tex]
 
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  • #2
Well, if the atom is known to be in the excited state, the probability that it you could measure it in the ground state is 0, right? If so, that means the two states are orthogonal. Similarly, if you know that the box is in the one-photon state, the probability of measuring zero photons in the box is 0, so those two states are orthogonal.
 
  • #3
KFC said:
I wonder if these two states [tex]|0, e\rangle[/tex] and [tex]|1\rangle|g\rangle[/tex] are orthorgonal? Why?
The definition of [itex]\langle 0,e|1,g\rangle[/itex] is

[tex]\langle 0,e|1,g\rangle=\langle e|\otimes\langle 0|\Big(|1\rangle\otimes|g\rangle\Big)=\langle 0|1\rangle\langle e|g\rangle[/tex]

and what diazona said explains why both factors on the right are zero.
 

FAQ: Product state and orthorgonality

What is product state in quantum mechanics?

Product state refers to a state in which a quantum system can be described as a direct product of two or more independent subsystems.

What is orthogonality in quantum mechanics?

Orthogonality refers to the property of two quantum states being perpendicular to each other in a mathematical sense. This means that the inner product of the two states is equal to zero.

How are product states and orthogonality related?

Product states and orthogonality are closely related in quantum mechanics. In fact, product states are always orthogonal to each other, meaning that their inner product is equal to zero. This is because product states represent independent subsystems that do not interact with each other.

What is the significance of orthogonality in quantum mechanics?

Orthogonality is a fundamental concept in quantum mechanics and has many important implications. It allows us to define a basis for a quantum system and provides a way to measure the differences between quantum states. It also plays a crucial role in quantum algorithms and quantum information processing.

How is orthogonality tested in experiments?

Orthogonality can be tested in experiments by measuring the inner product of two quantum states. This can be done using techniques such as quantum state tomography or quantum state discrimination. In some cases, orthogonality can also be inferred from other measurements, such as the overlap between two wavefunctions.

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