3 forgot a kronecker delta
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I would like to elaborate Adam Zalcman's answer, from a physical angle. What Adam actually showed is that all the eigenvalues of the time-evolution operator are of the form $e^{i\phi}$ where $\phi$ is real. The mathematical implication is that $U$ does not change the norm of states.

Let's look at the systems eigen-states, $\{|n\rangle\}$, which are defined by $$H|n\rangle=\epsilon_n|n\rangle$$ These states span the whole Hilbert space, so knowing how $U$ acts on them tells you everything you need to know about time evolution of an arbitrary state. Note that these states are also eigenvectors of $U$, because $$\begin{align} U(t)|n\rangle&=e^{\frac{i}{\hbar} H t}|n\rangle = \sum_k \frac{\left(\frac{i}{\hbar}Ht\right)^k}{k!}|n\rangle\\ & = \sum_k \frac{\left(\frac{i}{\hbar}\epsilon_n t\right) ^k }{k!}|n\rangle = e^{\frac{i}{\hbar} \epsilon_n t}|n\rangle \end{align}$$ and, indeed, each eigenvalue is of the form $e^{i\phi}$. This means physically that each eigenstate evolve in a very simple way - simply by changing its phase. An arbitrary state is of the form $$|\psi\rangle=\sum_n c_n |n\rangle$$ and its norm is $$\sqrt{\langle\psi|\psi\rangle}=\sqrt{\sum_{n,m}c_m^*c_n \langle m|n\rangle}=\sqrt{\sum_{nm}c_m^*c_n}=\sqrt{\sum_n |c_n|^2}$$$$\sqrt{\langle\psi|\psi\rangle}=\sqrt{\sum_{n,m}c_m^*c_n \langle m|n\rangle}=\sqrt{\sum_{nm}c_m^*c_n\delta_{mn}}=\sqrt{\sum_n |c_n|^2}$$ Since application of $U$ changes each $c_n$ only by its phase, it does not change the norm of $|\psi\rangle$.

BTW, since the absolute phase is not measurable, this implies that if the system is in a pure an eigenstate, it does not evolve in time. However, if the system is in a superposition of eigenstates, each eigenstate evolves with a different phase, according to their different energies, their relative phase changes and this is what causes stuff to change. Here's a nice applet that allows you to play with that.

I would like to elaborate Adam Zalcman's answer, from a physical angle. What Adam actually showed is that all the eigenvalues of the time-evolution operator are of the form $e^{i\phi}$ where $\phi$ is real. The mathematical implication is that $U$ does not change the norm of states.

Let's look at the systems eigen-states, $\{|n\rangle\}$, which are defined by $$H|n\rangle=\epsilon_n|n\rangle$$ These states span the whole Hilbert space, so knowing how $U$ acts on them tells you everything you need to know about time evolution of an arbitrary state. Note that these states are also eigenvectors of $U$, because $$\begin{align} U(t)|n\rangle&=e^{\frac{i}{\hbar} H t}|n\rangle = \sum_k \frac{\left(\frac{i}{\hbar}Ht\right)^k}{k!}|n\rangle\\ & = \sum_k \frac{\left(\frac{i}{\hbar}\epsilon_n t\right) ^k }{k!}|n\rangle = e^{\frac{i}{\hbar} \epsilon_n t}|n\rangle \end{align}$$ and, indeed, each eigenvalue is of the form $e^{i\phi}$. This means physically that each eigenstate evolve in a very simple way - simply by changing its phase. An arbitrary state is of the form $$|\psi\rangle=\sum_n c_n |n\rangle$$ and its norm is $$\sqrt{\langle\psi|\psi\rangle}=\sqrt{\sum_{n,m}c_m^*c_n \langle m|n\rangle}=\sqrt{\sum_{nm}c_m^*c_n}=\sqrt{\sum_n |c_n|^2}$$ Since application of $U$ changes each $c_n$ only by its phase, it does not change the norm of $|\psi\rangle$.

BTW, since the absolute phase is not measurable, this implies that if the system is in a pure an eigenstate, it does not evolve in time. However, if the system is in a superposition of eigenstates, each eigenstate evolves with a different phase, according to their different energies, their relative phase changes and this is what causes stuff to change. Here's a nice applet that allows you to play with that.

I would like to elaborate Adam Zalcman's answer, from a physical angle. What Adam actually showed is that all the eigenvalues of the time-evolution operator are of the form $e^{i\phi}$ where $\phi$ is real. The mathematical implication is that $U$ does not change the norm of states.

Let's look at the systems eigen-states, $\{|n\rangle\}$, which are defined by $$H|n\rangle=\epsilon_n|n\rangle$$ These states span the whole Hilbert space, so knowing how $U$ acts on them tells you everything you need to know about time evolution of an arbitrary state. Note that these states are also eigenvectors of $U$, because $$\begin{align} U(t)|n\rangle&=e^{\frac{i}{\hbar} H t}|n\rangle = \sum_k \frac{\left(\frac{i}{\hbar}Ht\right)^k}{k!}|n\rangle\\ & = \sum_k \frac{\left(\frac{i}{\hbar}\epsilon_n t\right) ^k }{k!}|n\rangle = e^{\frac{i}{\hbar} \epsilon_n t}|n\rangle \end{align}$$ and, indeed, each eigenvalue is of the form $e^{i\phi}$. This means physically that each eigenstate evolve in a very simple way - simply by changing its phase. An arbitrary state is of the form $$|\psi\rangle=\sum_n c_n |n\rangle$$ and its norm is $$\sqrt{\langle\psi|\psi\rangle}=\sqrt{\sum_{n,m}c_m^*c_n \langle m|n\rangle}=\sqrt{\sum_{nm}c_m^*c_n\delta_{mn}}=\sqrt{\sum_n |c_n|^2}$$ Since application of $U$ changes each $c_n$ only by its phase, it does not change the norm of $|\psi\rangle$.

BTW, since the absolute phase is not measurable, this implies that if the system is in a pure an eigenstate, it does not evolve in time. However, if the system is in a superposition of eigenstates, each eigenstate evolves with a different phase, according to their different energies, their relative phase changes and this is what causes stuff to change. Here's a nice applet that allows you to play with that.

2 typo: exponent k dropped out of formula
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I would like to elaborate Adam Zalcman's answer, from a physical angle. What Adam actually showed is that all the eigenvalues of the time-evolution operator are of the form $e^{i\phi}$ where $\phi$ is real. The mathematical implication is that $U$ does not change the norm of states.

Let's look at the systems eigen-states, $\{|n\rangle\}$, which are defined by $$H|n\rangle=\epsilon_n|n\rangle$$ These states span the whole Hilbert space, so knowing how $U$ acts on them tells you everything you need to know about time evolution of an arbitrary state. Note that these states are also eigenvectors of $U$, because $$\begin{align} U(t)|n\rangle&=e^{\frac{i}{\hbar} H t}|n\rangle = \sum_k \frac{\left(\frac{i}{\hbar}Ht\right)^k}{k!}|n\rangle\\ &=\sum_k \frac{\left(\frac{i}{\hbar}\epsilon_n t\right)}{k!}|n\rangle = e^{\frac{i}{\hbar} \epsilon_n t}|n\rangle \end{align}$$$$\begin{align} U(t)|n\rangle&=e^{\frac{i}{\hbar} H t}|n\rangle = \sum_k \frac{\left(\frac{i}{\hbar}Ht\right)^k}{k!}|n\rangle\\ & = \sum_k \frac{\left(\frac{i}{\hbar}\epsilon_n t\right) ^k }{k!}|n\rangle = e^{\frac{i}{\hbar} \epsilon_n t}|n\rangle \end{align}$$ and, indeed, each eigenvalue is of the form $e^{i\phi}$. This means physically that each eigenstate evolve in a very simple way - simply by changing its phase. An arbitrary state is of the form $$|\psi\rangle=\sum_n c_n |n\rangle$$ and its norm is $$\sqrt{\langle\psi|\psi\rangle}=\sqrt{\sum_{n,m}c_m^*c_n \langle m|n\rangle}=\sqrt{\sum_{nm}c_m^*c_n}=\sqrt{\sum_n |c_n|^2}$$ Since application of $U$ changes each $c_n$ only by its phase, it does not change the norm of $|\psi\rangle$.

BTW, since the absolute phase is not measurable, this implies that if the system is in a pure an eigenstate, it does not evolve in time. However, if the system is in a superposition of eigenstates, each eigenstate evolves with a different phase, according to their different energies, their relative phase changes and this is what causes stuff to change. Here's a nice applet that allows you to play with that.

I would like to elaborate Adam Zalcman's answer, from a physical angle. What Adam actually showed is that all the eigenvalues of the time-evolution operator are of the form $e^{i\phi}$ where $\phi$ is real. The mathematical implication is that $U$ does not change the norm of states.

Let's look at the systems eigen-states, $\{|n\rangle\}$, which are defined by $$H|n\rangle=\epsilon_n|n\rangle$$ These states span the whole Hilbert space, so knowing how $U$ acts on them tells you everything you need to know about time evolution of an arbitrary state. Note that these states are also eigenvectors of $U$, because $$\begin{align} U(t)|n\rangle&=e^{\frac{i}{\hbar} H t}|n\rangle = \sum_k \frac{\left(\frac{i}{\hbar}Ht\right)^k}{k!}|n\rangle\\ &=\sum_k \frac{\left(\frac{i}{\hbar}\epsilon_n t\right)}{k!}|n\rangle = e^{\frac{i}{\hbar} \epsilon_n t}|n\rangle \end{align}$$ and indeed each eigenvalue is of the form $e^{i\phi}$. This means physically that each eigenstate evolve in a very simple way - simply by changing its phase. An arbitrary state is of the form $$|\psi\rangle=\sum_n c_n |n\rangle$$ and its norm is $$\sqrt{\langle\psi|\psi\rangle}=\sqrt{\sum_{n,m}c_m^*c_n \langle m|n\rangle}=\sqrt{\sum_{nm}c_m^*c_n}=\sqrt{\sum_n |c_n|^2}$$ Since application of $U$ changes each $c_n$ only by its phase, it does not change the norm of $|\psi\rangle$.

BTW, since the absolute phase is not measurable, this implies that if the system is in a pure an eigenstate, it does not evolve in time. However, if the system is in a superposition of eigenstates, each eigenstate evolves with a different phase, according to their different energies, their relative phase changes and this is what causes stuff to change. Here's a nice applet that allows you to play with that.

I would like to elaborate Adam Zalcman's answer, from a physical angle. What Adam actually showed is that all the eigenvalues of the time-evolution operator are of the form $e^{i\phi}$ where $\phi$ is real. The mathematical implication is that $U$ does not change the norm of states.

Let's look at the systems eigen-states, $\{|n\rangle\}$, which are defined by $$H|n\rangle=\epsilon_n|n\rangle$$ These states span the whole Hilbert space, so knowing how $U$ acts on them tells you everything you need to know about time evolution of an arbitrary state. Note that these states are also eigenvectors of $U$, because $$\begin{align} U(t)|n\rangle&=e^{\frac{i}{\hbar} H t}|n\rangle = \sum_k \frac{\left(\frac{i}{\hbar}Ht\right)^k}{k!}|n\rangle\\ & = \sum_k \frac{\left(\frac{i}{\hbar}\epsilon_n t\right) ^k }{k!}|n\rangle = e^{\frac{i}{\hbar} \epsilon_n t}|n\rangle \end{align}$$ and, indeed, each eigenvalue is of the form $e^{i\phi}$. This means physically that each eigenstate evolve in a very simple way - simply by changing its phase. An arbitrary state is of the form $$|\psi\rangle=\sum_n c_n |n\rangle$$ and its norm is $$\sqrt{\langle\psi|\psi\rangle}=\sqrt{\sum_{n,m}c_m^*c_n \langle m|n\rangle}=\sqrt{\sum_{nm}c_m^*c_n}=\sqrt{\sum_n |c_n|^2}$$ Since application of $U$ changes each $c_n$ only by its phase, it does not change the norm of $|\psi\rangle$.

BTW, since the absolute phase is not measurable, this implies that if the system is in a pure an eigenstate, it does not evolve in time. However, if the system is in a superposition of eigenstates, each eigenstate evolves with a different phase, according to their different energies, their relative phase changes and this is what causes stuff to change. Here's a nice applet that allows you to play with that.

1
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I would like to elaborate Adam Zalcman's answer, from a physical angle. What Adam actually showed is that all the eigenvalues of the time-evolution operator are of the form $e^{i\phi}$ where $\phi$ is real. The mathematical implication is that $U$ does not change the norm of states.

Let's look at the systems eigen-states, $\{|n\rangle\}$, which are defined by $$H|n\rangle=\epsilon_n|n\rangle$$ These states span the whole Hilbert space, so knowing how $U$ acts on them tells you everything you need to know about time evolution of an arbitrary state. Note that these states are also eigenvectors of $U$, because $$\begin{align} U(t)|n\rangle&=e^{\frac{i}{\hbar} H t}|n\rangle = \sum_k \frac{\left(\frac{i}{\hbar}Ht\right)^k}{k!}|n\rangle\\ &=\sum_k \frac{\left(\frac{i}{\hbar}\epsilon_n t\right)}{k!}|n\rangle = e^{\frac{i}{\hbar} \epsilon_n t}|n\rangle \end{align}$$ and indeed each eigenvalue is of the form $e^{i\phi}$. This means physically that each eigenstate evolve in a very simple way - simply by changing its phase. An arbitrary state is of the form $$|\psi\rangle=\sum_n c_n |n\rangle$$ and its norm is $$\sqrt{\langle\psi|\psi\rangle}=\sqrt{\sum_{n,m}c_m^*c_n \langle m|n\rangle}=\sqrt{\sum_{nm}c_m^*c_n}=\sqrt{\sum_n |c_n|^2}$$ Since application of $U$ changes each $c_n$ only by its phase, it does not change the norm of $|\psi\rangle$.

BTW, since the absolute phase is not measurable, this implies that if the system is in a pure an eigenstate, it does not evolve in time. However, if the system is in a superposition of eigenstates, each eigenstate evolves with a different phase, according to their different energies, their relative phase changes and this is what causes stuff to change. Here's a nice applet that allows you to play with that.