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量子场论(四):实标量场产生湮灭算符的对易关系

2022-11-03 00:37 作者:我的世界-华汁  | 我要投稿


在三维空间对场算符%5Cphi(%5Cmathbf%20x%2Ct)做傅里叶变换:

%5Cint%5Cphi(x)e%5E%7Biq%5Ccdot%20x%7D%5Cmathrm%20d%5E3x%3D%5Cint%5Cfrac1%7B(2%5Cpi)%5E3%5Csqrt%7B2E_%5Cmathbf%20p%7D%7D%5Cbigg%5C%7B%5Cint%5Ba_%5Cmathbf%20p%20e%5E%7B-i(p-q)%5Ccdot%20x%7D%2Ba_%5Cmathbf%20p%5E%5Cdagger%20e%5E%7Bi(p%2Bq)%5Ccdot%20x%7D%5D%5Cmathrm%20d%5E3x%5Cbigg%5C%7D%5Cmathrm%20d%5E3p.%5Ctag%7B4.1%7D

由(2.36)式:

%5Cint%20e%5E%7B%5Cpm%20i(p-q)%5Ccdot%20x%7D%5Cmathrm%20d%5E3x%3D%5Cint%20e%5E%7B%5Cpm%20i(p%5E0-q%5E0)t%7De%5E%7B%5Cmp%20i(%5Cmathbf%20p-%5Cmathbf%20q)%5Ccdot%20%5Cmathbf%20x%7D%5Cmathrm%20d%5E3x%3D(2%5Cpi)%5E3e%5E%7B%5Cpm%20i(p%5E0-q%5E0)t%7D%5Cdelta%5E%7B(3)%7D(%5Cmathbf%20p-%5Cmathbf%20q).%5Ctag%7B4.2%7D

%5Cint%20e%5E%7B%5Cpm%20i(p%2Bq)%5Ccdot%20x%7D%5Cmathrm%20d%5E3x%3D%5Cint%20e%5E%7B%5Cpm%20i(p%5E0%2Bq%5E0)t%7De%5E%7B%5Cmp%20i(%5Cmathbf%20p%2B%5Cmathbf%20q)%5Ccdot%20%5Cmathbf%20x%7D%5Cmathrm%20d%5E3x%3D(2%5Cpi)%5E3e%5E%7B%5Cpm%20i(p%5E0%2Bq%5E0)t%7D%5Cdelta%5E%7B(3)%7D(%5Cmathbf%20p%2B%5Cmathbf%20q).%5Ctag%7B4.3%7D

于是得到:

%5Cint%5Cphi(x)e%5E%7Biq%5Ccdot%20x%7D%5Cmathrm%20d%5E3x%3D%5Cint%5Cfrac1%7B%5Csqrt%7B2E_%5Cmathbf%20p%7D%7D%5Ba_%5Cmathbf%20pe%5E%7B-i(p%5E0-q%5E0)t%7D%5Cdelta%5E%7B(3)%7D(%5Cmathbf%20p-%5Cmathbf%20q)%2Ba_%5Cmathbf%20p%5E%5Cdagger%20e%5E%7Bi(p%5E0%2Bq%5E0)t%7D%5Cdelta%5E%7B(3)%7D(%5Cmathbf%20p%2B%5Cmathbf%20q)%5D%5Cmathrm%20d%5E3p%5C%5C%3D%5Cfrac1%7B%5Csqrt%7B2E_%5Cmathbf%20q%7D%7D(a_%5Cmathbf%20q%2Ba_%7B-%5Cmathbf%20q%7D%5E%5Cdagger%20e%5E%7B2iq%5E0t%7D).%5Ctag%7B4.4%7D

共轭动量密度%5Cpi(%5Cmathbf%20x%2Ct)的傅里叶变换为:

%5Cint%5Cpi(x)e%5E%7Biq%5Ccdot%20x%7D%5Cmathrm%20d%5E3x%3D%5Cint%5Cfrac%7B-i%5Csqrt%7BE_%5Cmathbf%20p%7D%7D%7B%5Csqrt2(2%5Cpi)%5E3%7D%5Cbigg%5C%7B%5Cint%5Ba_%5Cmathbf%20pe%5E%7B-i(p-q)%5Ccdot%20x%7D-a_%5Cmathbf%20p%5E%5Cdagger%20e%5E%7Bi(p%2Bq)%5Ccdot%20x%7D%5D%5Cmathrm%20d%5E3x%5Cbigg%5C%7D%5Cmathrm%20d%5E3p%5C%5C%3D%5Cint%5Cfrac%7B-i%5Csqrt%7BE_%5Cmathbf%20p%7D%7D%7B%5Csqrt2%7D%5Ba_%5Cmathbf%20pe%5E%7B-i(p%5E0-q%5E0)t%7D%5Cdelta%5E%7B(3)%7D(%5Cmathbf%20p-%5Cmathbf%20q)-a%5E%5Cdagger_%5Cmathbf%20pe%5E%7Bi(p%5E0%2Bq%5E0)t%7D%5Cdelta%5E%7B(3)%7D(%5Cmathbf%20p%2B%5Cmathbf%20q)%5D%5Cmathrm%20d%5E3p%5C%5C%3D%5Cfrac%7B-i%5Csqrt%7BE_%5Cmathbf%20p%7D%7D%7B%5Csqrt2%7D(a_%5Cmathbf%20q-a_%7B-%5Cmathbf%20q%7D%5E%5Cdagger%20e%5E%7B2iq%5E0t%7D).%5Ctag%7B4.5%7D

从而:

%5Cint%5B%5Cpi(x)-iE_%5Cmathbf%20q%5Cphi(x)%5De%5E%7Biq%5Ccdot%20x%7D%5Cmathrm%20d%5E3x%3D-i%5Csqrt%7B2E_%5Cmathbf%20q%7Da_%5Cmathbf%20q.%5Ctag%7B4.6%7D

于是:

a_%5Cmathbf%20p%3D%5Cfrac%7Bi%7D%7B%5Csqrt%7B2E_%5Cmathbf%20p%7D%7D%5Cint%5B%5Cpi(x)-iE_%5Cmathbf%20p%5Cphi(x)%5De%5E%7Bip%5Ccdot%20x%7D%5Cmathrm%20d%5E3x.%5Ctag%7B4.7%7D

再于是(取厄米共轭):

a%5E%5Cdagger_%5Cmathbf%20p%3D%5Cfrac%7B-i%7D%7B%5Csqrt%7B2E_%5Cmathbf%20p%7D%7D%5Cint%5B%5Cpi(x)%2BiE_%5Cmathbf%20p%5Cphi(x)%5De%5E%7B-ip%5Ccdot%20x%7D%5Cmathrm%20d%5E3x.%5Ctag%7B4.8%7D

于是我们就可以推导产生湮灭算符的对易关系了。(小声

开始推导:

%5Cbegin%7Balign%7D%5Ba_%5Cmathbf%20p%2Ca%5E%5Cdagger_%5Cmathbf%20q%5D%26%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B4E_%5Cmathbf%20pE_%5Cmathbf%20q%7D%7D%5Cint%5B%5C%7B%5Cpi(%5Cmathbf%20x%2Ct)-iE_%5Cmathbf%20p%5Cphi(%5Cmathbf%20x%2Ct)%5C%7De%5E%7Bip%5Ccdot%20x%7D%2C%5C%7B%5Cpi(%5Cmathbf%20y%2Ct)%2BiE_%5Cmathbf%20q%5Cphi(%5Cmathbf%20y%2Ct)%5C%7De%5E%7B-iq%5Ccdot%20y%7D%5D%5Cmathrm%20d%5E3x%5Cmathrm%20d%5E3y%5C%5C%26%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B4E_%5Cmathbf%20pE_%5Cmathbf%20q%7D%7D%5Cint%5B%5Cpi(%5Cmathbf%20x%2Ct)-iE_%5Cmathbf%20p%5Cphi(%5Cmathbf%20x%2Ct)%2C%5Cpi(%5Cmathbf%20y%2Ct)%2BiE_%5Cmathbf%20q%5Cphi(%5Cmathbf%20y%2Ct)%5De%5E%7Bi(p%5Ccdot%20x-q%5Ccdot%20y)%7D%5Cmathrm%20d%5E3x%5Cmathrm%20d%5E3y%5C%5C%26%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B4E_%5Cmathbf%20pE_%5Cmathbf%20q%7D%7D%5Cint%5C%7BiE_%5Cmathbf%20q%5B%5Cpi(%5Cmathbf%20x%2Ct)%2C%5Cphi(%5Cmathbf%20y%2Ct)%5D-iE_%5Cmathbf%20p%5B%5Cphi(%5Cmathbf%20x%2Ct)%2C%5Cpi(%5Cmathbf%20y%2Ct)%5D%5C%7De%5E%7Bi(p%5E0-q%5E0)t%7De%5E%7B-i(%5Cmathbf%20p%5Ccdot%5Cmathbf%20x-%5Cmathbf%20q%5Ccdot%5Cmathbf%20y)%7D%5Cmathrm%20d%5E3x%5Cmathrm%20d%5E3y%5C%5C%26%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B4E_%5Cmathbf%20pE_%5Cmathbf%20q%7D%7D%5Cint%5B-i(E_%5Cmathbf%20p%2BE_%5Cmathbf%20q)i%5Cdelta%5E%7B(3)%7D(%5Cmathbf%20x-%5Cmathbf%20y)%5De%5E%7Bi(p%5E0-q%5E0)t%7De%5E%7B-i(%5Cmathbf%20p%5Ccdot%5Cmathbf%20x-%5Cmathbf%20q%5Ccdot%5Cmathbf%20y)%7D%5Cmathrm%20d%5E3x%5Cmathrm%20d%5E3y%5C%5C%26%3D%5Cfrac%7BE_%5Cmathbf%20p%2BE_%5Cmathbf%20q%7D%7B%5Csqrt%7B4E_%5Cmathbf%20pE_%5Cmathbf%20q%7D%7De%5E%7Bi(E_%5Cmathbf%20p-E_%5Cmathbf%20q)t%7D%5Cint%20e%5E%7B-i(%5Cmathbf%20p-%5Cmathbf%20q)%5Ccdot%20%5Cmathbf%20x%7D%5Cmathrm%20d%5E3x%5C%5C%26%3D%5Cfrac%7BE_%5Cmathbf%20p%2BE_%5Cmathbf%20q%7D%7B%5Csqrt%7B4E_%5Cmathbf%20pE_%5Cmathbf%20q%7D%7De%5E%7Bi(E_%5Cmathbf%20p-E_%5Cmathbf%20q)t%7D(2%5Cpi)%5E3%5Cdelta%5E%7B(3)%7D(%5Cmathbf%20p-%5Cmathbf%20q)%5C%5C%26%3D%5Cfrac%7BE_%5Cmathbf%20q%2BE_%5Cmathbf%20q%7D%7B%5Csqrt%7B4E_%5Cmathbf%20qE_%5Cmathbf%20q%7D%7De%5E%7Bi(E_%5Cmathbf%20q-E_%5Cmathbf%20q)t%7D(2%5Cpi)%5E3%5Cdelta%5E%7B(3)%7D(%5Cmathbf%20p-%5Cmathbf%20q)%5C%5C%26%3D(2%5Cpi)%5E3%5Cdelta%5E%7B(3)%7D(%5Cmathbf%20p-%5Cmathbf%20q).%5Cend%7Balign%7D%5Ctag%7B4.9%7D

类似地有:

%5Cbegin%7Balign%7D%5Ba_%5Cmathbf%20p%2Ca_%5Cmathbf%20q%5D%26%3D%5Cfrac%7B-1%7D%7B%5Csqrt%7B4E_%5Cmathbf%20pE_%5Cmathbf%20q%7D%7D%5Cint%5B%5Cpi(%5Cmathbf%20x%2Ct)-iE_%5Cmathbf%20p%5Cphi(%5Cmathbf%20x%2Ct)%2C%5Cpi(%5Cmathbf%20y%2Ct)-iE_%5Cmathbf%20q%5Cphi(%5Cmathbf%20y%2Ct)%5De%5E%7Bi(p%5Ccdot%20x%2Bq%5Ccdot%20y)%7D%5Cmathrm%20d%5E3x%5Cmathrm%20d%5E3y%5C%5C%26%3D%5Cfrac%7B-1%7D%7B%5Csqrt%7B4E_%5Cmathbf%20pE_%5Cmathbf%20q%7D%7D%5Cint%5C%7B-iE_%5Cmathbf%20q%5B%5Cpi(%5Cmathbf%20x%2Ct)%2C%5Cphi(%5Cmathbf%20y%2Ct)%5D-iE_%5Cmathbf%20p%5B%5Cphi(%5Cmathbf%20x%2Ct)%2C%5Cpi(%5Cmathbf%20y%2Ct)%5D%5C%7De%5E%7Bi(p%5E0%2Bq%5E0)t%7De%5E%7B-i(%5Cmathbf%20p%5Ccdot%5Cmathbf%20x%2B%5Cmathbf%20q%5Ccdot%5Cmathbf%20y)%7D%5Cmathrm%20d%5E3x%5Cmathrm%20d%5E3y%5C%5C%26%3D%5Cfrac%7B-1%7D%7B%5Csqrt%7B4E_%5Cmathbf%20pE_%5Cmathbf%20q%7D%7D%5Cint%20e%5E%7Bi(p%5E0%2Bq%5E0)t%7De%5E%7B-i(%5Cmathbf%20p%5Ccdot%5Cmathbf%20x%2B%5Cmathbf%20q%5Ccdot%5Cmathbf%20y)%7D(E_%5Cmathbf%20p-E_%5Cmathbf%20q)%5Cdelta%5E%7B(3)%7D(%5Cmathbf%20x-%5Cmathbf%20y)%5Cmathrm%20d%5E3x%5Cmathrm%20d%5E3y%5C%5C%26%3D%5Cfrac%7BE_%5Cmathbf%20q-E_%5Cmathbf%20p%7D%7B%5Csqrt%7B4E_%5Cmathbf%20pE_%5Cmathbf%20q%7D%7De%5E%7Bi(E_%5Cmathbf%20p%2BE_%5Cmathbf%20q)t%7D%5Cint%20e%5E%7B-i(%5Cmathbf%20p%2B%5Cmathbf%20q)%5Ccdot%20%5Cmathbf%20x%7D%5Cmathrm%20d%5E3x%5C%5C%26%3D%5Cfrac%7BE_%5Cmathbf%20q-E_%5Cmathbf%20p%7D%7B%5Csqrt%7B4E_%5Cmathbf%20pE_%5Cmathbf%20q%7D%7De%5E%7Bi(E_%5Cmathbf%20p%2BE_%5Cmathbf%20q)t%7D(2%5Cpi)%5E3%5Cdelta%5E%7B(3)%7D(%5Cmathbf%20p%2B%5Cmathbf%20q)%5C%5C%26%3D0.%5Cend%7Balign%7D%5Ctag%7B4.10%7D

而且:

%5Ba_%5Cmathbf%20p%5E%5Cdagger%2Ca_%5Cmathbf%20q%5E%5Cdagger%5D%3D%5Ba_%5Cmathbf%20p%2Ca_%5Cmathbf%20q%5D%5E%5Cdagger%3D0.%5Ctag%7B4.11%7D

综上,产生湮灭算符的对易关系为:

%5Ba_%5Cmathbf%20p%2Ca%5E%5Cdagger_%5Cmathbf%20q%5D%3D(2%5Cpi)%5E3%5Cdelta%5E%7B(3)%7D(%5Cmathbf%20p-%5Cmathbf%20q)%5C%20%2C%5C%20%5Ba_%5Cmathbf%20p%2Ca_%5Cmathbf%20q%5D%3D%5Ba_%5Cmathbf%20p%5E%5Cdagger%2Ca_%5Cmathbf%20q%5E%5Cdagger%5D%3D0.%5Ctag%7B4.12%7D

量子场论(四):实标量场产生湮灭算符的对易关系的评论 (共 条)

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