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## Main Question or Discussion Point

I have some soubts on quantizying theories..how many techniques are there?..is always as simple as to take p=-ihd/dx ?..or the functional derivative?..thanks.

- Thread starter eljose79
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I have some soubts on quantizying theories..how many techniques are there?..is always as simple as to take p=-ihd/dx ?..or the functional derivative?..thanks.

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selfAdjoint

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Anyway, the "p=-id/dx" you refer to is called a 'representation' of the operator on a 'representation space'. The representation space you have picked (representing the Hilbert space mentioned above) is the space of wavefunctions (in particular, the "coordinate rep"), and in this rep, the operator, p, takes the form you wrote abovem while the operator x is represented as multiplication by the number x.

You could have chosen the 'momentum representation', meaning your rep space is the space of (wave)functions of momentum rather than position. Then the operator p would be represented by multiplication by p, while the operator x would be of the form d/dp.

The wavefunction rep of qm is what Schrodinger used. You could instead use the "state vector" representation (used in the intro QM text of Liboff and by Feynman in his lectures in physics, vol 3). This is how Heisenberg formulated qm.

All of this stuff does not represent what we would call different methods of quantization, rather just different representations of the canonical quantization of non-relativistic point particles.

The issue of different methods of quantization becomes more significant in doing *relativistic* quantum mechanics where we must talk about quantum field theory (QFT). There, it turns out that Feynman's "path integral quantization" is more suitable, but to see why you have to study QFT.

Cheers

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selfAdjoint

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