Relation Between the Wave Function and Probability Density
The quantity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Psi(\textbf{r},t)|^2} can be interpreted as probability density. In order for us to do so, two conditions must be met. First, the probability amplitude must be positive semi-definite (equal to or greater than zero). This condition is trivial because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Psi(\textbf{r},t)|^2 \!} is always a non-negative function. Second, the probability density, integrated over all space, must be equal to one:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{-\infty}^{\infty}d^3\textbf{r}\,|\Psi(\textbf{r},t)|^2=1}
We will show that, if this relation is satisfied for a specific time, then it is satisfied for all times shortly.
Because of the fact that we may interpret Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Psi(\textbf{r},t)|^2} as a probability density, we may calculate expectation values of observables, such as position and momentum, in terms of it. In general, the expectation value of an observable Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q(\textbf{r},\textbf{p};t)} is given by
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle Q(\textbf{r},\textbf{p};t)\rangle=\int_{-\infty}^{\infty}d^3\textbf{r}\,\Psi^{\ast}(\textbf{r},t)Q\left (\textbf{r},-i\hbar\nabla;t\right )\Psi(\textbf{r},t).}
In particular, the expectation value of a position coordinate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i} is
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle x_i\rangle=\int_{-\infty}^{\infty}d^3\textbf{r}\,x_i|\Psi(\textbf{r},t)|^2}
and that for a component of momentum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_i} is
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle p_i\rangle=\int_{-\infty}^{\infty}d^3\textbf{r}\,\Psi^{\ast}(\textbf{r},t)\left (-i\hbar\frac{\partial}{\partial x_i}\right )\Psi(\textbf{r},t).}
Conservation of Probability
We will now show that the solution to the Schrödinger equation conserves probability, i.e. the probability to find the particle somewhere in the space does not change with time. To see that it does, consider
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i\hbar\frac{\partial}{\partial t}\Psi(\textbf{r},t)=\left[-\frac{\hbar^2}{2m}\nabla^2+V(\textbf{r})\right]\Psi(\textbf{r},t)}
Now multiply both sides by the complex conjugate of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(\textbf{r},t):\!}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i\hbar\Psi^{\ast}(\textbf{r},t)\frac{\partial}{\partial t}\Psi(\textbf{r},t)=\Psi^{\ast}(\textbf{r},t)\left[-\frac{\hbar^2}{2m}\nabla^2+V(\textbf{r})\right]\Psi(\textbf{r},t)}
Now, take the complex conjugate of this entire expression:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -i\hbar\Psi(\textbf{r},t)\frac{\partial}{\partial t}\Psi^{\ast}(\textbf{r},t)=\Psi(\textbf{r},t)\left[-\frac{\hbar^2}{2m}\nabla^2+V(\textbf{r})\right]\Psi^{\ast}(\textbf{r},t)}
Taking the difference of the above equations, we finally find
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial}{\partial t} \left( \Psi^{\ast}(\textbf{r},t)\Psi(\textbf{r},t)\right) + \frac{\hbar}{2im} \nabla \cdot \left[\Psi^{\ast}(\textbf{r},t)\nabla \Psi(\textbf{r},t)-(\nabla\Psi^{\ast}(\textbf{r},t)) \Psi(\textbf{r},t)\right]=0}
Note that this is in the form of a continuity equation,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial}{\partial t} \rho(\textbf{r},t) + \nabla \cdot \textbf{j}(\textbf{r},t)=0,}
where
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho(\textbf{r},t)=\Psi^{\ast}(\textbf{r},t)\Psi(\textbf{r},t)\!}
is the probability density, and
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textbf{j}(\textbf{r},t)=-\frac{i\hbar}{2m}\left[\Psi^{\ast}(\textbf{r},t)\nabla \Psi(\textbf{r},t)-(\nabla\Psi^{\ast}(\textbf{r},t)) \Psi(\textbf{r},t)\right]}
is the probability current.
Once we know that the densities and currents constructed from the solution of the Schrödinger equation satisfy the continuity equation, it is easy to show that the probability is conserved.
To see this, note that
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dt}\int d^3\textbf{r} |\Psi(\textbf{r},t)|^2=-\int d^3\textbf{r}(\nabla\cdot \textbf{j})=-\oint d\textbf{A} \cdot \textbf{j} =0.}
Here, we used the fact that the wave function is assumed to vanish outside of the boundary, and thus the current vanishes as well. Therefore, we see that the normalization of the wave function does not change over time, so that we only need to normalize it at one instant in time, as asserted earlier.
Problems
(1) Consider a particle moving in a potential field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(\textbf{r}).}
(a) Prove that the average energy is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle E\rangle=\int W\,d^3\textbf{r}=\int\left (\frac{\hbar^2}{2m}\nabla\Psi^{\ast}\cdot\nabla\Psi+\Psi^{\ast}V\Psi\right )\,d^3\textbf{r},} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} is energy density.
(b) Prove the energy conservation equation, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial W}{\partial t}+\nabla \cdot \textbf{S}=0,} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textbf{S}=-\frac{\hbar^2}{2m}\left (\frac{\partial\Psi^{\ast}}{\partial t}\nabla\Psi + \frac{\partial\Psi}{\partial t}\nabla\Psi^{\ast}\right )} is the energy flux density.
(2) Assume that the Hamiltonian for a system of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N}
particles is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{H}=-\sum_{i=1}^{N}\frac{\hbar}{2m}\nabla_{i}^{2}+\sum_{i=1}^{N}\rho_{ij}(|\textbf{r}_{i}-\textbf{r}_{j}|)}
, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi(\textbf{r}_{1},\textbf{r}_{2},\ldots,\textbf{r}_{N};t)}
is the wave fuction. Defining
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho(\textbf{r},t)=\sum\rho_{i}(\textbf{r},t),}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textbf{j}(\textbf{r},t)=\sum\textbf{j}_{i}(\textbf{r},t),}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_{1}(\textbf{r}_{1},t)=\int\cdots\int d^{3}\textbf{r}_{2}\,d^{3}\textbf{r}_{3}\,\cdots\,d^{3}\textbf{r}_{N}\,\Psi^{\star}\Psi,}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textbf{j}_{1}(\textbf{r}_{1},t)=-\frac{i\hbar}{2m}\int\cdots\int d^{3}\textbf{r}_{2}\,d^{3}\textbf{r}_{3}\,\cdots\,d^{3}\textbf{r}_{N}\,(\Psi^{\star}\nabla_{1}\Psi-\Psi\nabla_{1}\Psi^{\star}),}
and similarly for the other Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_i} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textbf{j}_i} , prove the following relation:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial\rho}{\partial t}+\nabla\cdot\textbf{j}=0}