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Dirac left me in the dust at this step. Can someone please help me through it.
I understand from Susskind's lectures that if we need a constant, that we can assign any label to it, and that i[itex]\hbar[/itex] is merely a prescient choice. But I fail to understand Dirac's logic of why "[itex]\hbar[/itex] must simply be a number."
Also, am I right in feeling that HUP just appeared out of nowhere in this mysterious algebraic step? If so, Dirac didn't call attention to it in this chapter.
p.s. I apologize for a previously fumbled attempt to post this question.
P.M. Dirac said:[u[itex]_{1}[/itex],v[itex]_{1}[/itex]](u[itex]_{2}[/itex]v[itex]_{2}[/itex]-v[itex]_{2}[/itex]u[itex]_{2}[/itex])=[u[itex]_{2}[/itex],v[itex]_{2}[/itex]](u[itex]_{1}[/itex]v[itex]_{1}[/itex]-v[itex]_{1}[/itex]u[itex]_{1}[/itex])
Since this condition holds with u[itex]_{1}[/itex] and v[itex]_{1}[/itex] quite independent of u[itex]_{2}[/itex] and v[itex]_{2}[/itex], we must have.
u[itex]_{1}[/itex]v[itex]_{1}[/itex]-v[itex]_{1}[/itex]u[itex]_{1}[/itex] = -i[itex]\hbar[/itex][u[itex]_{1}[/itex],v[itex]_{1}[/itex]]
u[itex]_{2}[/itex]v[itex]_{2}[/itex]-v[itex]_{2}[/itex]u[itex]_{2}[/itex] = -i[itex]\hbar[/itex][u[itex]_{2}[/itex],v[itex]_{2}[/itex]]
where [itex]\hbar[/itex] must not depend on u[itex]_{1}[/itex] and v[itex]_{1}[/itex], nor on u[itex]_{2}[/itex] and v[itex]_{2}[/itex] and also must commute with u[itex]_{1}[/itex]v[itex]_{1}[/itex]-v[itex]_{1}[/itex]u[itex]_{1}[/itex]. It follows that [itex]\hbar[/itex] must simply be a number.
...
We are thus led to the following definition for the quantum P.B. [u,v] of any two variables u and v.
uv-vu = i[itex]\hbar[/itex][u.v]
I understand from Susskind's lectures that if we need a constant, that we can assign any label to it, and that i[itex]\hbar[/itex] is merely a prescient choice. But I fail to understand Dirac's logic of why "[itex]\hbar[/itex] must simply be a number."
Also, am I right in feeling that HUP just appeared out of nowhere in this mysterious algebraic step? If so, Dirac didn't call attention to it in this chapter.
p.s. I apologize for a previously fumbled attempt to post this question.