Quantum Spectrum of a Hamiltonian with Linear Potential

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The Hamiltonian with a linear potential, represented as z x, leads to Airy function solutions when considering a particle in a 1-D infinite potential well, resulting in numerical eigenvalues. This behavior is confirmed by simulations, such as those conducted with Mathematica, that illustrate the effects of a stationary electric field on an electron in confinement. In cases where the potential well has finite walls, obtaining a numerical solution becomes significantly more complex. The derivation of these concepts can be found in David Miller's Quantum book. Understanding the spectrum of such Hamiltonians is crucial for quantum mechanics applications.
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Suppose to have a hamiltonian with a linear potenzial like z x, where x is the variable and z a parameter. Which is the spectrum of the Hamiltonian of this sistem?
 
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Depends... if the particle is to be bounded in a 1-D infinite potential well then a linearly-varying potential gives rise to Airy function solutions and hence the eigenvalues are inherently numerical (I remember this distinctly from a recent Mathematica simulation of a stationary electric field perturbing an electron confined to a 1-D box). You can find a derivation in David Miller's Quantum book. If the walls are finite then the numerical solution is more difficult to find.
 

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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