# Lower bounds on energy eigenvalues

## Main Question or Discussion Point

Hi,
I'm interested in learning about what would be the compliment to the Variational method. I'm aware that the Variational method allows one to calculate upper bounds, but I'm wondering about methods to calculate lower bounds on energy eigenvalues. And for energies besides the ground state if such methods exist.

Are there methods to calculate lower bounds on energy eigenvalues (the ground state and higher)?

Thanks,

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ShayanJ
Gold Member
The lower bound for an energy spectrum is provided by the Uncertainty principle. You should just be able to use it properly. I think with a little thought, you can use it to find the lower bound for any state but it can't be generally formulated and is different for each case.
About finding the energy upper bound for states other than the ground state. For the ground state, you just guess(it can be an educated guess) a wave function and calculate the Hamiltonian's expectation value in that state and minimize it.
For the first excited state, you should again guess a wave function, but this time, you have the condition that this wave function should be orthogonal to the one you chose for the ground state. Then you do the same thing as the last case.
For second excited state, the wave function you guess should be orthogonal to the last two wave functions and so on.

The lower bound for an energy spectrum is provided by the Uncertainty principle. You should just be able to use it properly. I think with a little thought, you can use it to find the lower bound for any state but it can't be generally formulated and is different for each case.
About finding the energy upper bound for states other than the ground state. For the ground state, you just guess(it can be an educated guess) a wave function and calculate the Hamiltonian's expectation value in that state and minimize it.
For the first excited state, you should again guess a wave function, but this time, you have the condition that this wave function should be orthogonal to the one you chose for the ground state. Then you do the same thing as the last case.
For second excited state, the wave function you guess should be orthogonal to the last two wave functions and so on.
Thanks for your response. You already taught me something! However, I was thinking more along the lines of something like Temple's inequality. An inequality relation using expectation values of the Hamiltonian. I have been trying to find something similar for other energies besides the ground state.

ShayanJ
Gold Member
I don't know about that inequality so I can't help.