Return to Answer

Stick to one dimension, as the identification only holds for one space dimension, so you might well call it a coincidence, $$ j={\hbar\over 2mi}(\psi^*\partial_x \psi -\psi \partial_x \psi^*), \\ \partial_t \rho + \partial_x j =0 ~~~~\leadsto ~~~ v(x) = j/\rho ~~. $$

Vanishing Wronskian suggests (not quite implies) that $\psi^*$ is linearly dependent on $\psi$, so they may only differ by a constant phase, absorbable, so the wave function is basically real, and the probability density constant over time. This is what you get in stationary systems.

A constant current J also indicates, for $\psi\equiv \sqrt{\rho} ~ e^{is(x)}$, a special velocity $$ J={\hbar\rho \over m} \partial_x s, ~~~~\leadsto ~~~ \psi\equiv \sqrt{\rho} ~~e^{i(\alpha+ \int^x \!dx' ~ Jm/\hbar \rho(x') ) } $$ so ρ is again time independent, even though ψ is genuinely complex.

Stick to one dimension, as the identification only holds for one space dimension, so you might well call it a coincidence, $$ j={\hbar\over 2mi}(\psi^*\partial_x \psi -\psi \partial_x \psi^*), \\ \partial_t \rho + \partial_x j =0 ~~~~\leadsto ~~~ v(x) = j/\rho ~~. $$

Vanishing Wronskian suggests (not quite dictates) that $\psi^*$ is linearly dependent on $\psi$, i.e. $\partial_x (\ln (\psi^*/\psi))=0$; so they may only differ by a constant phase, absorbable, hence the wave function is basically real, and the probability density constant over time. This is what you get in stationary systems.

A constant current J also indicates, for $\psi\equiv \sqrt{\rho} ~ e^{is(x)}$, a special velocity $$ J={\hbar\rho \over m} \partial_x s, ~~~~\leadsto ~~~ \psi\equiv \sqrt{\rho} ~~e^{i(\alpha+ \int^x \!dx' ~ Jm/\hbar \rho(x') ) } $$ so ρ is again time independent, even though ψ is genuinely complex.

Stick to one dimension, as the identification only holds for one space dimension, so you might well call it a coincidence, $$ j={\hbar\over 2mi}(\psi^*\partial_x \psi -\psi \partial_x \psi^*), \\ \partial_t \rho + \partial_x j =0 ~~~~\leadsto ~~~ v(x) = j/\rho ~~. $$

Vanishing Wronskian implies that $\psi^*$ is linearly dependent on $\psi$, so they may only differ by a constant phase, absorbable, so the wave function is basically real, and the probability density constant over time. This is what you get in stationary systems.

A constant current J also indicates, for $\psi\equiv \sqrt{\rho} ~ e^{is(x)}$, a special velocity $$ J={\hbar\rho \over m} \partial_x s, ~~~~\leadsto ~~~ \psi\equiv \sqrt{\rho} ~~e^{i(\alpha+ \int^x \!dx' ~ Jm/\hbar \rho(x') ) } $$ so ρ is again time independent, even though ψ is genuinely complex.

Stick to one dimension, as the identification only holds for one space dimension, so you might well call it a coincidence, $$ j={\hbar\over 2mi}(\psi^*\partial_x \psi -\psi \partial_x \psi^*), \\ \partial_t \rho + \partial_x j =0 ~~~~\leadsto ~~~ v(x) = j/\rho ~~. $$

Vanishing Wronskian suggests (not quite implies) that $\psi^*$ is linearly dependent on $\psi$, so they may only differ by a constant phase, absorbable, so the wave function is basically real, and the probability density constant over time. This is what you get in stationary systems.

A constant current J also indicates, for $\psi\equiv \sqrt{\rho} ~ e^{is(x)}$, a special velocity $$ J={\hbar\rho \over m} \partial_x s, ~~~~\leadsto ~~~ \psi\equiv \sqrt{\rho} ~~e^{i(\alpha+ \int^x \!dx' ~ Jm/\hbar \rho(x') ) } $$ so ρ is again time independent, even though ψ is genuinely complex.

Stick to one dimension, as the identification only holds for one space dimension, $$ j={\hbar\over 2mi}(\psi^*\partial_x \psi -\psi \partial_x \psi^*), \\ \partial_t \rho + \partial_x j =0 ~~~~\leadsto ~~~ v(x) = j/\rho ~~. $$

Vanishing Wronskian implies that $\psi^*$ is linearly dependent on $\psi$, so they may only differ by a constant phase, absorbable, so the wave function is basically real, and the probability density constant over time. This is what you get in stationary systems.

A constant current J also indicates, for $\psi\equiv \sqrt{\rho} ~ e^{is(x)}$, a special velocity $$ J={\hbar\rho \over m} \partial_x s, ~~~~\leadsto ~~~ \psi\equiv \sqrt{\rho} ~~e^{i(\alpha+ \int^x \!dx' ~ Jm/\hbar \rho(x') ) } $$ so ρ is again time independent, even though ψ is genuinely complex.

Stick to one dimension, as the identification only holds for one space dimension, so you might well call it a coincidence, $$ j={\hbar\over 2mi}(\psi^*\partial_x \psi -\psi \partial_x \psi^*), \\ \partial_t \rho + \partial_x j =0 ~~~~\leadsto ~~~ v(x) = j/\rho ~~. $$

Vanishing Wronskian implies that $\psi^*$ is linearly dependent on $\psi$, so they may only differ by a constant phase, absorbable, so the wave function is basically real, and the probability density constant over time. This is what you get in stationary systems.

A constant current J also indicates, for $\psi\equiv \sqrt{\rho} ~ e^{is(x)}$, a special velocity $$ J={\hbar\rho \over m} \partial_x s, ~~~~\leadsto ~~~ \psi\equiv \sqrt{\rho} ~~e^{i(\alpha+ \int^x \!dx' ~ Jm/\hbar \rho(x') ) } $$ so ρ is again time independent, even though ψ is genuinely complex.