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higgsss
higgsss
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Hermiticity of $H$ for all three cases combined together can be shown directly from the Schrödinger equation. To do this, first take the derivative of the unitarity relation $U(t,t_{0}) U^{\dagger}(t,t_{0}) = I$ with respect to $t$, which gives \begin{equation} U(t,t_{0}) \left[\frac{\partial}{\partial t}U^{\dagger}(t,t_{0}) \right] = -\left[\frac{\partial}{\partial t}U(t,t_{0}) \right]U^{\dagger}(t,t_{0}). \end{equation} Then, consider the Hermitian conjugate of \begin{equation} H = i\hbar \left[\frac{\partial}{\partial t}U(t,t_{0}) \right] U^{\dagger}(t,t_{0}), \end{equation} which reads \begin{equation} \begin{split} H^{\dagger} &= -i\hbar\, U(t,t_{0})\left[\frac{\partial}{\partial t}U^{\dagger}(t,t_{0}) \right] \\ &= i\hbar\, \left[\frac{\partial}{\partial t}U(t,t_{0}) \right]U^{\dagger}(t,t_{0}). \end{split} \end{equation} Therefore, $H = H^\dagger$, and hence $H$ is Hermitian.

Hermiticity of $H$ can be shown directly from the Schrödinger equation. To do this, first take the derivative of the unitarity relation $U(t,t_{0}) U^{\dagger}(t,t_{0}) = I$ with respect to $t$, which gives \begin{equation} U(t,t_{0}) \left[\frac{\partial}{\partial t}U^{\dagger}(t,t_{0}) \right] = -\left[\frac{\partial}{\partial t}U(t,t_{0}) \right]U^{\dagger}(t,t_{0}). \end{equation} Then, consider the Hermitian conjugate of \begin{equation} H = i\hbar \left[\frac{\partial}{\partial t}U(t,t_{0}) \right] U^{\dagger}(t,t_{0}), \end{equation} which reads \begin{equation} \begin{split} H^{\dagger} &= -i\hbar\, U(t,t_{0})\left[\frac{\partial}{\partial t}U^{\dagger}(t,t_{0}) \right] \\ &= i\hbar\, \left[\frac{\partial}{\partial t}U(t,t_{0}) \right]U^{\dagger}(t,t_{0}). \end{split} \end{equation} Therefore, $H = H^\dagger$, and hence $H$ is Hermitian.

Hermiticity of $H$ for all three cases combined together can be shown directly from the Schrödinger equation. To do this, first take the derivative of the unitarity relation $U(t,t_{0}) U^{\dagger}(t,t_{0}) = I$ with respect to $t$, which gives \begin{equation} U(t,t_{0}) \left[\frac{\partial}{\partial t}U^{\dagger}(t,t_{0}) \right] = -\left[\frac{\partial}{\partial t}U(t,t_{0}) \right]U^{\dagger}(t,t_{0}). \end{equation} Then, consider the Hermitian conjugate of \begin{equation} H = i\hbar \left[\frac{\partial}{\partial t}U(t,t_{0}) \right] U^{\dagger}(t,t_{0}), \end{equation} which reads \begin{equation} \begin{split} H^{\dagger} &= -i\hbar\, U(t,t_{0})\left[\frac{\partial}{\partial t}U^{\dagger}(t,t_{0}) \right] \\ &= i\hbar\, \left[\frac{\partial}{\partial t}U(t,t_{0}) \right]U^{\dagger}(t,t_{0}). \end{split} \end{equation} Therefore, $H = H^\dagger$, and hence $H$ is Hermitian.

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higgsss
higgsss
  • 2.9k
  • 2
  • 17
  • 26

Hermiticity of $H$ can be shown directly from the Schrödinger equation. To do this, first take the derivative of the unitarity relation $U(t,t_{0}) U^{\dagger}(t,t_{0}) = I$ with respect to $t$, which gives \begin{equation} U(t,t_{0}) \left[\frac{\partial}{\partial t}U^{\dagger}(t,t_{0}) \right] = -\left[\frac{\partial}{\partial t}U(t,t_{0}) \right]U^{\dagger}(t,t_{0}). \end{equation} Then, consider the Hermitian conjugate of \begin{equation} H = i\hbar \left[\frac{\partial}{\partial t}U(t,t_{0}) \right] U^{\dagger}(t,t_{0}), \end{equation} which reads \begin{equation} \begin{split} H^{\dagger} &= -i\hbar\, U(t,t_{0})\left[\frac{\partial}{\partial t}U^{\dagger}(t,t_{0}) \right] \\ &= i\hbar\, \left[\frac{\partial}{\partial t}U(t,t_{0}) \right]U^{\dagger}(t,t_{0}). \end{split} \end{equation} Therefore, $H = H^\dagger$, and hence $H$ is Hermitian.