Undergrad Is the Dimensionality of Vector Spaces the Same for Different Quantum States?

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The discussion highlights the difference in dimensionality between energy eigenbasis and position eigenbasis in quantum mechanics, using the particle in a box problem as an example. It notes that the energy eigenbasis is countably infinite, while the position eigenbasis is uncountably infinite due to the continuous range of values in the interval [0, L]. The dimensionality of the vector space is only the same if the eigenstates belong to the same topological vector space. It concludes that the space of eigenvectors for position (X) is larger than that for energy (H), indicating that their spectral equations do not share solutions in the same space. This distinction is crucial for understanding the nature of quantum states and their representations.
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Consider the particle in a box problem. The number of energy eigenbasis is 'countable' infinity. But the number of position eigenbasis is 'uncountable' infinity. x can take any value from the interval [0,L] Whichever basis I choose, shouldn't the dimensionality of the vector space be the same?
 
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That is true, iff the „eigenstates” are element of the same topological vector space. But the space of the eigenvectors of X is larger than the space of the eigenvectors of H, or, equivalently, the two spectral equations for X and H do not have solutions in the same space.
 
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Thread 'Two equivalent statements of time reversal symmetric Hamiltonian'

Time reversal invariant Hamiltonians must satisfy ##[H,\Theta]=0## where ##\Theta## is time reversal operator. However, in some texts (for example see Many-body Quantum Theory in Condensed Matter Physics an introduction, HENRIK BRUUS and KARSTEN FLENSBERG, Corrected version: 14 January 2016, section 7.1.4) the time reversal invariant condition is introduced as ##H=H^*##. How these two conditions are identical?
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