1st order pertubation on 2 level system

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SUMMARY

The discussion centers on first-order perturbation theory applied to a two-level quantum system with different parity. The first-order energy corrections for both energy levels are confirmed to be zero due to the parity condition, as expressed by E11=<ψ1| H' |ψ1>. However, the first-order wavefunction corrections are non-zero, calculated using the formula ψ11=<ψ2| H' | ψ1>/(E1-E2) ψ2. This highlights the distinction between energy corrections and wavefunction corrections in quantum mechanics.

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luxiaolei
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Hi all. I m stucked on the question followed, any helps will be greatly appreciated.

A perturbation has the form H'=z act on a two level system which they have different parity.

So the first order correction to the energy level 1 and 2 are give by:

E11=<ψ1| H' |ψ1>

Same for level 2.

These two energy correction are clearly zero because the same level has same parity.

However the first order wavefunction corrections are not for both level:

According to time independent non-degenerate perturbation theory on the 1st order correction to the wave function formula:

ψ11=<ψ2| H ' | ψ1>/(E1-E2) ψ2

Which is none zero

Where am I wrong? I am so confused..thanks in advance.
 
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Your understanding is correct. The first-order energy corrections for both levels are zero, since they have the same parity and H' has the form z. However, the first-order wavefunction corrections are not zero, as you have calculated correctly.
 

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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