Proof of normalization constant of wave function to be independent of time

Question

I am trying to prove that the normalization constant is independent of time. If we have fixed it for a particular time then it will remain constant for all time.

Suppose $\psi(x,t)$ is a wavefunction.
Let $A(t)$ be the normalization constant of $\psi(x,t)$
Then $\displaystyle A^*A\int_{-\infty}^{\infty}|\psi(x,t)|^2dx=1\tag{1}$
$\displaystyle \implies \frac{d}{dt}A^*A\int_{-\infty}^{\infty}|\psi(x,t)|^2dx=0\tag{2}$
$\displaystyle \implies\int_{-\infty}^{\infty}|\psi(x,t)|^2dx\frac{d}{dt}A^*A+A^*A\frac{d}{dt}\int_{-\infty}^{\infty}|\psi(x,t)|^2dx=0\tag{3}$

Now first analyze, $\displaystyle \frac{d}{dt}\int_{-\infty}^{\infty}|\psi(x,t)|^2dx$
$\displaystyle \frac{d}{dt}\int_{-\infty}^{\infty}|\psi(x,t)|^2dx=\int_{-\infty}^{\infty}\frac{\partial}{\partial t}|\psi(x,t)|^2dx\tag{4}$
$\displaystyle \frac{\partial}{\partial t}|\psi(x,t)|^2dx=\frac{\partial}{\partial t}(\psi^*\psi)dx=\psi^*\frac{\partial\psi}{\partial t}=\psi\frac{\partial\psi^*}{\partial t}\tag{5}$
By time dependent Schrodinger equation
$\displaystyle \frac{\partial\psi}{\partial t}=\frac{i\bar{h}}{2m}\frac{\partial^2\psi}{\partial x^2}-\frac{i}{\bar{h}}V\psi\tag{6}$
Also, $\displaystyle \frac{\partial\psi^*}{\partial t}=-\frac{i\bar{h}}{2m}\frac{\partial^2\psi^*}{\partial x^2}+\frac{i}{\bar{h}}V\psi^*\tag{7}$

So, $\displaystyle \frac{\partial|\psi|^2}{\partial t}=\psi^*\Big(\frac{i\bar{h}}{2m}\frac{\partial^2\psi}{\partial x^2}-\frac{i}{\bar{h}}V\psi\Big)+\psi\Big(-\frac{i\bar{h}}{2m}\frac{\partial^2\psi^*}{\partial x^2}+\frac{i}{\bar{h}}V\psi^*\Big)\tag{8}$
$\displaystyle \implies\frac{\partial|\psi|^2}{\partial t}=\frac{i\bar{h}}{2m}\Big(\frac{\partial^2\psi}{\partial x^2}-\frac{\partial^2\psi^*}{\partial x^2}\Big)\tag{9}$
$\displaystyle \implies\frac{\partial|\psi|^2}{\partial t}=\frac{\partial}{\partial x}\Big[\frac{i\bar{h}}{2m}\Big(\psi^*\frac{\partial\psi}{\partial x}-\psi\frac{\partial\psi^*}{\partial x}\Big)\Big]\tag{10}$
$\displaystyle \frac{d}{dt}\int_{-\infty}^{\infty}|\psi(x,t)|^2dx=\int_{-\infty}^{\infty}\frac{\partial}{\partial x}\Big[\frac{i\bar{h}}{2m}\Big(\psi^*\frac{\partial\psi}{\partial x}-\psi\frac{\partial\psi^*}{\partial x}\Big)\Big]dx\tag{11}$
$\displaystyle \implies\frac{d}{dt}\int_{-\infty}^{\infty}|\psi(x,t)|^2dx=\frac{i\bar{h}}{2m}\Big(\psi^*\frac{\partial\psi}{\partial x}-\psi\frac{\partial\psi^*}{\partial x}\Big)\Big|_{-\infty}^\infty\tag{12}$
As $\displaystyle \psi(x,t)\to0$ as $x\to\pm\infty$.
So, $\displaystyle \frac{d}{dt}\int_{-\infty}^{\infty}|\psi(x,t)|^2dx=0\tag{13}$
So, $(3)$ becomes
$\displaystyle \implies\int_{-\infty}^{\infty}|\psi(x,t)|^2dx\frac{d}{dt}A^*A=0\tag{14}$
As $\psi$ is square integrable, so $\int_{-\infty}^{\infty}|\psi(x,t)|^2dx=c$ where $c\in\mathbb R$ and $c\neq 0$
So, $\frac{d}{dt}A^*A=0\tag{15}$
$\implies |A|^2=constant\tag{16}$

I have the following doubts from the proof
(i) From $(11)$ to $(12)$, in the RHS, we have used fundamental theorem of calculus.
Ingtegrating of the derivative is the antiderivative. $\int_a^bf'(x)dx=f(x)$. But here the condition is that f has to continuous and differentiable on $[a,b]$ with $f'$ integrable on $[a,b]$.
So, in $\int_{-\infty}^{\infty}\frac{\partial}{\partial x}\Big[\frac{i\bar{h}}{2m}\Big(\psi^*\frac{\partial\psi}{\partial x}-\psi\frac{\partial\psi^*}{\partial x}\Big)\Big]dx$, we take $\Big(\psi^*\frac{\partial\psi}{\partial x}-\psi\frac{\partial\psi^*}{\partial x}\Big)$ to be continuous. As $\psi$ and $\psi^*$ is continuous, this means that $\frac{\partial\psi}{\partial x}$ is also continuous. But we know in general that first derivative of $\psi$ can be discontinuous also. So how we have used fundamental theorem of calculus here?

(ii) From $(12)$ to $(13)$, we have taken $\frac{i\bar{h}}{2m}\Big(\psi^*\frac{\partial\psi}{\partial x}-\psi\frac{\partial\psi^*}{\partial x}\Big)\Big|_{-\infty}^\infty=0$ using the fact that $\psi\to 0$ as $x\to\pm\infty$. But this also means that $\frac{\partial\psi}{\partial x}\to 0$ as $x\to\pm\infty$. But how we can be sure that $\frac{\partial\psi}{\partial x}$ is bounded?

(iii) In $(16)$, we get $|A|^2=constant$. But from this how we get $A(t)=constant$.
$|A|^2$ constant means that magnitude of the vector in complex plane is constant but it might happen that the angle of $A$ changes. This angle changes as a function of $t$. So how we conclude that $A$ is independent of time?

Answer 1

To make this completely rigorous, you would have to delve into functional analysis. But I will try to explain it intuitively.

(i) Arguably all derivatives of $\psi$ are continuous in reality. Non-smooth functions are an idealization, but a useful one. They can be formalized using distributions. Even when $f$ is discontinuous, $f'$ is defined to be something that integrates to $f$. The results are equivalent to considering smooth functions and taking a limit in which they develop a discontinuity or kink.

(ii) The wave function must have a real expectation value of kinetic energy on physical grounds. The kinetic energy operator is proportional to $-\partial^2/\partial x^2$ and has an expectation value proportional to $$\int_{-\infty}^\infty dx\, \psi^* \left(-\frac{\partial^2\psi}{\partial x^2}\right) = \int_{-\infty}^\infty dx\, \left|\frac{\partial\psi}{\partial x}\right|^2 - \left.\psi^* \frac{\partial\psi}{\partial x}\right|_{-\infty}^\infty.$$ The imaginary part of this is precisely proportional to your key expression, $(\psi^*\, \partial\psi/\partial x - \psi\, \partial\psi^*/\partial x)|_{-\infty}^\infty$, which must therefore be zero. (Technically, the wave function must be defined in a space where the energy operator is Hermitian.)

(iii) The Schrödinger equation fully determines the evolution of the wave function $\psi$, and what you have shown is that if $\psi$ is initially normalized ($|A| = 1$) then it remains normalized ($|A|$ remains $1$). By saying that $A$ is the normalization constant, you have only defined its magnitude. So indeed the phase of $A$ is arbitrary, but it is not part of the wave function. Given that $|A| = 1$, you can simply write the normalization condition as $\int_{-\infty}^\infty dx\, |\psi|^2 = 1$. Ultimately, $A$ is a redundant, unphysical parameter.

Answer 2

If the Hamiltonian is time-independent, then it seems much simpler to use $$ \vert\Psi(t)\rangle = e^{-i \hat H t/\hbar}\vert\Psi(0)\rangle\, . $$ Then it is automatic that \begin{align} \langle \Psi(t)\vert\Psi(t)\rangle &= \langle \Psi(0)\vert e^{i\hat H t/\hbar} e^{-i \hat H t/\hbar}\vert \Psi(0)\rangle\, ,\\ &=\langle \Psi(0)\vert \Psi(0)\rangle \end{align} since $e^{i\hat H t/\hbar}e^{-i \hat H t/\hbar}=\mathbb{1}$.

If the Hamiltonian is time dependent, then the evolution operator $U(t)$ is constructed to be unitary, so that \begin{align} \vert\Psi(t)\rangle &= U(t)\vert\Psi(0)\rangle \, ,\\ \langle \Psi(0)\vert U^\dagger(t)U(t)\vert \Psi(0)\rangle&= \langle \Psi(0)\vert \Psi(0)\rangle \end{align} as $U^\dagger(t)U=\mathbb{1}$ by construction.

The wavefunction version follows in the same way: \begin{align} \Psi(x,t)&=e^{-i \hat H t/\hbar}\Psi(x,0)\, ,\\ \Psi(x,t)&=U(t)\Psi(x,0) \end{align} for the time-independent and the time-dependent cases, respectively.

Answer 3

Regarding (i), I am still not sure what you are asking. In general, $d\psi/dx$ is continuous over an interval if the potential $V(x)$ is finite within that interval. At the boundaries, if the potential $V(x) \to \pm \infty$, then $d\psi/dx$ is no longer continuous. Therefore, the continuity of the first derivative of the wavefunction depends on the behavior of the potential.

(See, for example, Griffiths' Introduction to Quantum Mechanics Chapter 2, Section 2.5.2 The Delta-Function Well.)

Regarding (ii), for a physical system, the wavefunctions "decays" to zero as one approaches $\pm \infty$ on the real position axis -- otherwise the system is unphysical. This means the wavefunction eventually becomes flat, implying its first derivative is zero.

Regarding (iii), you are correct in that $A$ might have some "angular" dependence. This is called phase dependence and is mostly inconsequential in Quantum Mechanics. You can always isolate the phase as follows: $A(t) = |A| \, e^{i\, \phi(t)}$. After that, once you evaluate $|A|^2 = A(t)^* A(t)$, the phase factor reduces to unity.