Question

2. Schrodinger equation In quantum mechanics, physical quantities correspond to Hermitian operators. In particular, the total energy of the system corresponds to the Hamiltonian operator $H$, which is a hermitian operator. The 'state of the system' is a time dependent vector in an inner product space, $|psi(t) angle$. The state of the system obeys the Schrodinger equation $ihbarfrac{d}{dt}|psi(t) angle = H|psi(t) angle$. We assume that there are no time-varying external forces on the system, so that the Hamiltonian operator $H$ is not itself time-dependent. a) Take the Hermitian conjugate of the Schrodinger equation, which puts it into bra form (this is a simple one-liner; use the fact that $H$ is hermitian). b) Show that $langlepsi(t)|psi(t) angle$ is time independent, i.e. $frac{d}{dt}langlepsi(t)|psi(t) angle = 0$. (hint: use the product rule). This means that we can normalize the state vector at one time, say $t = 0$, so that $langlepsi(0)|psi(0) angle = 1$, and it will stay normalized at all times, $langlepsi(t)|psi(t) angle = 1$. c) Define the time-evolution operator $U(t) = e^{-iHt/hbar}$ where the exponential is defined by a Taylor expansion in powers of $H$: $U(t) = I - frac{it}{hbar}H + frac{1}{2}(frac{-it}{hbar})^2H^2 + ...$ Show that $U(t)$ is unitary: $U(t)^dagger U(t) = U(t)U(t)^dagger = I$. d) Let $f(H)$ be any function of $H$ that can be defined by a Taylor expansion in powers of $H$. Show that $H$ and $f(H)$ commute. e) Find an expression for $frac{d}{dt}U(t)$. f) Show that $|psi(t) angle = U(t)|psi(0) angle$ is a solution of the Schrodinger equation. Here $|psi(0) angle$ is the initial state of the system, which we assume is given.

          2. Schrodinger equation In quantum mechanics, physical quantities correspond to Hermitian operators. In particular, the total energy of the system corresponds to the Hamiltonian operator $H$, which is a hermitian operator. The 'state of the system' is a time dependent vector in an inner product space, $|psi(t)
angle$. The state of the system obeys the Schrodinger equation
$ihbarfrac{d}{dt}|psi(t)
angle = H|psi(t)
angle$.
We assume that there are no time-varying external forces on the system, so that the Hamiltonian operator $H$ is not itself time-dependent.
a) Take the Hermitian conjugate of the Schrodinger equation, which puts it into bra form (this is a simple one-liner; use the fact that $H$ is hermitian).
b) Show that $langlepsi(t)|psi(t)
angle$ is time independent, i.e.
$frac{d}{dt}langlepsi(t)|psi(t)
angle = 0$.
(hint: use the product rule). This means that we can normalize the state vector at one time, say $t = 0$, so that $langlepsi(0)|psi(0)
angle = 1$, and it will stay normalized at all times, $langlepsi(t)|psi(t)
angle = 1$.
c) Define the time-evolution operator
$U(t) = e^{-iHt/hbar}$
where the exponential is defined by a Taylor expansion in powers of $H$:
$U(t) = I - frac{it}{hbar}H + frac{1}{2}(frac{-it}{hbar})^2H^2 + ...$
Show that $U(t)$ is unitary: $U(t)^dagger U(t) = U(t)U(t)^dagger = I$.
d) Let $f(H)$ be any function of $H$ that can be defined by a Taylor expansion in powers of $H$. Show that $H$ and $f(H)$ commute.
e) Find an expression for $frac{d}{dt}U(t)$.
f) Show that $|psi(t)
angle = U(t)|psi(0)
angle$ is a solution of the Schrodinger equation. Here $|psi(0)
angle$ is the initial state of the system, which we assume is given.

$2. Schrodinger equation In quantum mechanics, physical quantities correspond to Hermitian operators. In particular, the total energy of the system corresponds to the Hamiltonian operator H, which is a hermitian operator. The 'state of the system' is a time dependent vector in an inner product space, |psi(t) angle. The state of the system obeys the Schrodinger equation ihbarfracddt|psi(t) angle = H|psi(t) angle. We assume that there are no time-varying external forces on the system, so that the Hamiltonian operator H is not itself time-dependent. a) Take the Hermitian conjugate of the Schrodinger equation, which puts it into bra form (this is a simple one-liner; use the fact that H is hermitian). b) Show that langlepsi(t)|psi(t) angle is time independent, i.e. fracddtlanglepsi(t)|psi(t) angle = 0. (hint: use the product rule). This means that we can normalize the state vector at one time, say t = 0, so that langlepsi(0)|psi(0) angle = 1, and it will stay normalized at all times, langlepsi(t)|psi(t) angle = 1. c) Define the time-evolution operator U(t) = e^-iHt/hbar where the exponential is defined by a Taylor expansion in powers of H: U(t) = I - fracithbarH + frac12(frac-ithbar)^2H^2 + ... Show that U(t) is unitary: U(t)^dagger U(t) = U(t)U(t)^dagger = I. d) Let f(H) be any function of H that can be defined by a Taylor expansion in powers of H. Show that H and f(H) commute. e) Find an expression for fracddtU(t). f) Show that |psi(t) angle = U(t)|psi(0) angle is a solution of the Schrodinger equation. Here |psi(0) angle is the initial state of the system, which we assume is given.$

Added by Iker T.

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