Equivalent representations of stationary states in Quantum Mechanics

Question

The time-dependent Schrodinger equation is given as $$i \hbar \frac{\mathrm d}{\mathrm dt}| \psi(t) \rangle = \hat{H} | \psi(t) \rangle. $$ To find how the states evolve in time we want to find the linear operator $\hat{U}(t,t_0)$ such that $$| \psi(t) \rangle = \hat{U}(t,t_0)| \psi(t_0) \rangle.$$ Substituting into Schrodinger equation yields $$\frac{\partial \hat{U}(t,t_0)}{ \partial t} = - \frac{i}{\hbar}\hat{H}\hat{U}(t,t_0)$$ this leads to $$\hat{U}(t,t_0) = e^{\frac{-i(t-t_0)\hat{H}}{\hbar}}$$ hence $$| \psi(t) \rangle = e^{-\frac{i(t-t_0)\hat{H}}{\hbar}}|\psi(t_0) \rangle$$ where $e^{a \hat{A}} = \sum \frac{a^n}{n !}\hat{A}^n = \hat{I} + a \hat{A} + \frac{a^2}{2 !} \hat{A}^2 + \frac{a^3}{3 !}\hat{A}^3 +...$

Question:
Given the operator series expansion of $e^{a \hat{A}}$ above, how does it follow that if we consider a time independent Hamiltonian $\hat{H_0}$ where solutions- the eigenvalues $E_n$ and eigenstates $| \psi_n \rangle$ (stationary states)- are equivalently stated as $$e^{\frac{-it \hat{H_0}}{\hbar}}| \psi_n \rangle = e^{-\frac{i E_n t}{\hbar}} | \psi_n \rangle?$$

Thanks.

Answer 1

If $A$ is an arbitrary operator, with eigenbasis $$ A|a\rangle=a|a\rangle\tag{1} $$ then the operator $f(A)$ is defined through $$ f(A)|a\rangle=f(a)|a\rangle\tag{2} $$

If the function $f$ is analytic, $$ f(x)=\sum_n \frac{f^{(n)}(0)}{n!}x^n\tag{3} $$ then you can also define $$ f(A)\equiv \sum_n \frac{f^{(n)}(0)}{n!}A^n\tag{4} $$ and, provided the sum converges, it agrees with $(2)$, as it follows from $A^n|a\rangle=a^n|a\rangle$ (here you have to assume that the Taylor series of $f$ converges for all $x$ in the spectrum of $A$).

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In your particular case, $A=H$, $f=\exp$, and $|a\rangle=|\psi_n\rangle$, which means that $$ \mathrm e^{iHt}|\psi_n\rangle=\mathrm e^{-iE_nt}|\psi_n\rangle \tag{5} $$ holds by definition (cf. $(2)$).