교환 연산자

양자역학에서 교환 연산자(exchange operator) 또는 바꿈 연산자란 두 동일한 입자의 라벨을 바꾸는 연산자이다. 예를 들어, 두 개의 동일한 입자를 라벨 1,2를 붙여 구별하고 이 입자로 기술되는 의 파동함수를 uEσ1σ2(1,2)로 쓰자. 그리고 이에 대한 교환연산자를 P1,2라 쓰면 이 연산자의 역할은 다음을 말한다.

운동상수

퍼텐셜스핀과 관련이 없는 경우, 교환연산자는 운동상수이다. 이를 확인하기 위해 간단한 두 개의 동일한 입자로 이루어진 계에서 해밀토니안 연산자와 교환연산자에 교환법칙이 성립하는지 알아보자.

먼저, 스핀퍼텐셜이 관련이 없는 경우, 해밀토니언과 퍼텐셜은 아래와 같이 쓸 수 있다.

위 해밀토니언을 입자의 라벨에 대한 함수로 만들어

라 쓸 수 있다. 그러면, 퍼텐셜이 입자의 교환에 대해 대칭이기 때문에 아래와 같이 해밀토니안 또한 대칭이된다.

이 해밀토니언을 파동함수에 작용시키면

이다. 그리고 해밀토니언의 대칭성에 의해

을 얻는다. 위의 교환 연산자의 역할을 다시 쓰면,

이다. 이제 두 연산자에 대해 교환법칙이 성립하는지 알아보기 위해 두 연산자를 동시에 파동 함수에 작용시켜보자.

즉, 두 연산자 사이에 교환법칙이 성립하므로,

이다. 따라서, 교환연산자는 이 경우 운동상수가 된다.

고윳값과 고유상태

반전성과 유사하게 교환연산자도 두 번 연속으로 연산을 하게 되면, 원래의 상태로 돌아오게 된다.

따라서 P1,2=1 이므로 고윳값은 ±1 이 돤다.

고유상태 또한 반전성처럼 우함수기함수고유상태인 것처럼 +1인 경우 대칭조합된 파동함수가, -1인 경우 반대칭조합된 파동함수가 고유상태가 된다.

고윳값 +1 :
고윳값 -1 :

여기서 각 N들은 규격화상수이다.

같이 보기

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