Definition of Two-Level Systems in Quantum Mechanics

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A "two-level system" in quantum mechanics refers to a quantum state represented as a linear combination of two basis states, typically denoted as |1> and |2>. The state vector can be expressed as |Ψ>=a|1>+b|2>, where the coefficients a and b represent the probability amplitudes. This concept is crucial for understanding systems with quantized energy levels, such as a particle in a box, where energy states are discrete (E1 and E2) while position remains continuous. The definition emphasizes that the classification as a two-level system is contingent upon the choice of basis, specifically the eigenstates of energy.

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JK423
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May you please help me with the following...
In quantum mechanics, what`s the definition of the "two-level systems"? I understand that the state vector is in the form |Ψ>=a|1>+b|2>, where |1>,|2> is a basis of the state space.
Then i think of the particle in a box. The energy is quantized (lets say that the possible values are E1 and E2) while position x is continuous. So, in the first case we would have: |Ψ>=a|E1>+b|E2> and in the second one: |Ψ>=Integral(Ψ(x) |x> dx).
So if we use as a basis the eigenstates of the energy, our system would be a "two-level system". However, in {x} representation, we would have an "infinite-level system".

So what`s the definition of a "two-level system" since the number of levels depend on the basis we use?
 
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