There is a particularly neat way to derive Schr\u00f6dinger’s equation, and to justify the “canonical substitution” rules for replacing energy and momentum with corresponding operators when we “quantize” an equation.<\/p>\n
Take a particle in a potential. Its energy is given by<\/p>\n
$$E=\\frac{{\\bf p}^2}{2m}+V({\\bf x}),$$<\/p>\n
or<\/p>\n
$$E-\\frac{{\\bf p}^2}{2m}-V({\\bf x})=0.$$<\/p>\n
Now multiply both sides this equation by the formula \\(e^{i({\\bf p}\\cdot{\\bf x}-Et)\/\\hbar}\\). We note that this exponential expression cannot ever be zero if the part in the exponent that’s in parentheses is real:<\/p>\n
$$\\left[E-\\frac{{\\bf p}^2}{2m}-V({\\bf x})\\right]e^{i({\\bf p}\\cdot{\\bf x}-Et)\/\\hbar}=0.$$<\/p>\n
So far so good. But now note that<\/p>\n
$$Ee^{i({\\bf p}\\cdot{\\bf x}-Et)\/\\hbar}=i\\hbar\\frac{\\partial}{\\partial t}e^{i({\\bf p}\\cdot{\\bf x}-Et)\/\\hbar},$$<\/p>\n
and similarly,<\/p>\n
$${\\bf p}^2e^{i({\\bf p}\\cdot{\\bf x}-Et)\/\\hbar}=-\\hbar^2{\\boldsymbol\\nabla}e^{i({\\bf p}\\cdot{\\bf x}-Et)\/\\hbar}.$$<\/p>\n
This allows us to rewrite the previous equation as<\/p>\n
$$\\left[i\\hbar\\frac{\\partial}{\\partial t}+\\hbar^2\\frac{{\\boldsymbol\\nabla}^2}{2m}-V({\\bf x})\\right]e^{i({\\bf p}\\cdot{\\bf x}-Et)\/\\hbar}=0.$$<\/p>\n
Or, writing \\(\\Psi=e^{i({\\bf p}\\cdot{\\bf x}-Et)\/\\hbar}\\) and rearranging:<\/p>\n
$$i\\hbar\\frac{\\partial}{\\partial t}\\Psi=-\\hbar^2\\frac{{\\boldsymbol\\nabla}^2}{2m}\\Psi+V({\\bf x})\\Psi,$$<\/p>\n
which is the good old Schr\u00f6dinger equation.<\/p>\n
The method works for an arbitrary, generic Hamiltonian, too. Given<\/p>\n
$$H({\\bf p})=E,$$<\/p>\n
we can write<\/p>\n
$$\\left[E-H({\\bf p})\\right]e^{i({\\bf p}\\cdot{\\bf x}-Et)\/\\hbar}=0,$$<\/p>\n
which is equivalent to<\/p>\n
$$\\left[i\\hbar\\frac{\\partial}{\\partial t}-H(-i\\hbar{\\boldsymbol\\nabla})\\right]\\Psi=0.$$<\/p>\n
So if this equation is identically satisfied for a classical system with Hamiltonian \\(H\\), what’s the big deal about quantum mechanics? Well… a classical system satisfies \\(E-H({\\bf p})=0\\), where \\(E\\) and \\({\\bf p}\\) are eigenvalues of the differential operators \\(i\\hbar\\partial\/\\partial t\\) and \\(-i\\hbar{\\boldsymbol\\nabla}\\), respectively. Schr\u00f6dinger’s equation, on the other hand, remains valid\u00a0in the general case, not just for the eigenvalues.<\/p>\n There is a particularly neat way to derive Schr\u00f6dinger’s equation, and to justify the “canonical substitution” rules for replacing energy and momentum with corresponding operators when we “quantize” an equation. Take a particle in a potential. Its energy is given by $$E=\\frac{{\\bf p}^2}{2m}+V({\\bf x}),$$ or $$E-\\frac{{\\bf p}^2}{2m}-V({\\bf x})=0.$$ Now multiply both sides this equation by […]<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[],"class_list":["post-6903","post","type-post","status-publish","format-standard","hentry","category-physics","category-3-id","post-seq-1","post-parity-odd","meta-position-corners","fix"],"_links":{"self":[{"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=\/wp\/v2\/posts\/6903","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=6903"}],"version-history":[{"count":17,"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=\/wp\/v2\/posts\/6903\/revisions"}],"predecessor-version":[{"id":6920,"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=\/wp\/v2\/posts\/6903\/revisions\/6920"}],"wp:attachment":[{"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6903"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6903"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6903"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}