양자역학 미분연산자 행렬표현

<p>미적분 기본정리가 코시-입실론 델타로 </p><p>수학적 논리성을 가진다.</p><p>즉, 이를 활용해서 미분연산자를</p><p>행렬로 표현해보자. </p><p>결과론적으로 아래와같다.</p><p>참고로 h는 미적분 극한변수이지,</p><p>플랑크상수아님.</p><div class="figure-img" data-ke-type="image" data-ke-style="alignCenter" data-ke-mobilestyle="widthOrigin"><img src="https://t1.daumcdn.net/cafeattach/1YOll/1f94009e1d2607cebd549d64645899d25341dce9" class="txc-image" data-img-src="https://t1.daumcdn.net/cafeattach/1YOll/1f94009e1d2607cebd549d64645899d25341dce9" data-origin-width="1672" data-origin-height="941"></div><p>증명해보자</p><div class="figure-img" data-ke-type="image" data-ke-style="alignCenter" data-ke-mobilestyle="widthOrigin"><img src="https://t1.daumcdn.net/cafeattach/1YOll/32e1ffe6d433c76f99888ccca78aa7690eb8761c" class="txc-image" data-img-src="https://t1.daumcdn.net/cafeattach/1YOll/32e1ffe6d433c76f99888ccca78aa7690eb8761c" data-origin-width="1672" data-origin-height="941"></div><p>여기서 전개하면</p><div class="figure-img" data-ke-type="image" data-ke-style="alignCenter" data-ke-mobilestyle="widthOrigin"><img src="https://t1.daumcdn.net/cafeattach/1YOll/cf874cc299cf35637b775afb9972e70fb2c6149e" class="txc-image" data-img-src="https://t1.daumcdn.net/cafeattach/1YOll/cf874cc299cf35637b775afb9972e70fb2c6149e" data-origin-width="1672" data-origin-height="941"></div><p>공백 행렬값을 추가해서 다시적으면</p><div class="figure-img" data-ke-type="image" data-ke-style="alignCenter" data-ke-mobilestyle="widthOrigin"><img src="https://t1.daumcdn.net/cafeattach/1YOll/bd8cc612fe19616a92303e6c00dc3a317ff1c945" class="txc-image" data-img-src="https://t1.daumcdn.net/cafeattach/1YOll/bd8cc612fe19616a92303e6c00dc3a317ff1c945" data-origin-width="1672" data-origin-height="941"></div><p>행렬로 나타내자.</p><div class="figure-img" data-ke-type="image" data-ke-style="alignCenter" data-ke-mobilestyle="widthOrigin"><img src="https://t1.daumcdn.net/cafeattach/1YOll/4c8a56de5ebaa4225c652ba51c57237deb2a043d" class="txc-image" data-img-src="https://t1.daumcdn.net/cafeattach/1YOll/4c8a56de5ebaa4225c652ba51c57237deb2a043d" data-origin-width="1672" data-origin-height="941"></div><p>미분으로 d/dx를 밖으로 빼면</p><div class="figure-img" data-ke-type="image" data-ke-style="alignCenter" data-ke-mobilestyle="widthOrigin"><img src="https://t1.daumcdn.net/cafeattach/1YOll/dcd8b634796e8a894e3bd21c16141b3680416782" class="txc-image" data-img-src="https://t1.daumcdn.net/cafeattach/1YOll/dcd8b634796e8a894e3bd21c16141b3680416782" data-origin-width="1672" data-origin-height="941"></div><p>그럼 최종적으로 연산자 부분만 쓰면</p><div class="figure-img" data-ke-type="image" data-ke-style="alignCenter" data-ke-mobilestyle="widthOrigin"><img src="https://t1.daumcdn.net/cafeattach/1YOll/b8ffb09c3f91e4c8679dc9cac2b3d26c6cd43db8" class="txc-image" data-img-src="https://t1.daumcdn.net/cafeattach/1YOll/b8ffb09c3f91e4c8679dc9cac2b3d26c6cd43db8" data-origin-width="1672" data-origin-height="941"></div><p>마지막으로</p><div class="figure-img" data-ke-type="image" data-ke-style="alignCenter" data-ke-mobilestyle="widthOrigin"><img src="https://t1.daumcdn.net/cafeattach/1YOll/9a97aa24e5819a2bf17363cd1d26e37c80ce6039" class="txc-image" data-img-src="https://t1.daumcdn.net/cafeattach/1YOll/9a97aa24e5819a2bf17363cd1d26e37c80ce6039" data-origin-width="1080" data-origin-height="924"></div><p>즉, 미분연산자는 Anti-Hermitian이고</p><p>운동량 연산자는 Hermitian이다. </p><p>모든 측정가능 연산자는 </p><p>Hermitian이어야하고 고유값은 실수이다. </p><p> </p><p>1. 연산자를 미분식으로 표현이 가능한 경우.</p><p>→ 해당변수  공간 연속</p><p>ex) x가 연속인 p나 각도(파이)가 연속인 L_z </p><p> </p><p>2. 연산자를 미분식이 아닌 유한행렬식으로만 표현되는경우 → 해당변수 공간 불연속</p><p>ex) 파울리 스핀연산자 </p><p> </p><p>양자역학을 들어가기에 앞서</p><p>미적분 기본정리와</p><p>선형대수(행렬)기본 법칙은 양자역학의</p><p>abc이다. </p>