Bound State Condition: Definition Explained

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The bound state condition is defined as having energy E < 0 and a wavefunction that decays to zero as r approaches infinity. However, there is debate regarding the definition, particularly in cases like the harmonic oscillator, which has positive energy yet still exhibits bound states. The discussion highlights that while wavefunctions must decay at infinity to satisfy the L2 condition, not all decaying wavefunctions correspond to bound states. Bound states can exist in finite potential wells or attractive delta functions, indicating that the condition is more nuanced than simply energy and decay. The conversation emphasizes the complexity of defining bound states in quantum mechanics.
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What is the precise definition of the bound state condition? Thanks in advance.
 
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E < 0 and that the wavefunction "decays" to 0 as r-> infty
 
malawi_glenn said:
E < 0 and that the wavefunction "decays" to 0 as r-> infty
Disagree with you, though I'm not insisting. Consider a harmonic oscillator: all energies > 0, still bound states. Wave function should "decay" at infinity, but something must be said about when it decays: e.g. a moving Gaussian wave packet decays faster than an exponent, but does not correspond to a bound state.
 
Well, a harmonic oscillator is bound by definition though. It's in an infinite potential well.

If it wasn't an infinite potential well, the particle could tunnel out sooner or later. So it wouldn't be bound then.
 
Yes, "If it wasn't an infinite potential well, the particle could tunnel out sooner or later. So it wouldn't be bound then".
 
Doesn't any allowed wave function decay to zero at infinity as part of the L2 condition? But not all wave functions are bound states, and we do have bound states in finite wells or attractive delta functions, so the bound state condition can't be just that.
Or am I missing the point, maybe?
 

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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