While going through Shankar's Principles of Quantum Mechanics (2nd edition), in the Chapter 5 at page 179, while explaining the concept of quantization of energy for particle in a box he argued (though I may be wrong) that we need equal amount of parameters and constraints to have a solution. I want to understand why do we need so. Also what if there are more parameters than given constraints? I am also attaching the picture of the said page.
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I think you simply reading the page wrong! It says: "we impose four conditions on $\psi$, which has only three free parameters . . .the continuity conditions cannot be fulfilled except possibly at certain special energies. That is the origin of energy quantization here." So if the numbers are equal we can just solve the equation for any eigenvalue, and if the number of constraints is smaller then there are multiple independent solutions. – Aug 09 '22 at 10:43
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Please edit your question to include the text that you are concerned with, rather than a screenshot. Including the image rather than the written-out text means that this question is not readable by users who use screen readers and similar technology. – Michael Seifert Aug 09 '22 at 14:05
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@CWPP thanks for showing my mistake. But it will be very kind of you if you explain me how equal numbers can just solve the equation for any eigenvalue and how the other scenario can happen. – Soumyadeep Chakraborty Aug 09 '22 at 18:45
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@MichaelSeifert I am really sorry for the trouble. But I post the picture along with the written question. I attached the picture as it is highly likely that I may have understood the whole concept wrong and thus may have framed a wrong question (which I have written too). Thanks for your guidance and I will be more more careful hereafter. – Soumyadeep Chakraborty Aug 09 '22 at 18:48