05 - Probability

Probability Density Function

Let $\Psi (\mathbf{x},t)$ be particle’s wave function

Probability Density Function for position of particle is

$$\rho (\mathbf{x},t)\equiv |\Psi (\mathbf{x},t){|}^2$$

Probability of Particle in Region

Let $\Psi (\mathbf{x},t)$ be the particle’s wave function

Probability of finding particle in volume $D\subset {\mathbb{R}}^3$ is

$${\mathbb{P}}_{\Psi }(D)={\iiint }_D|\Psi (\mathbf{x},t){|}^2{d}^{3}x$$

Normalisable Wave Function

Wave Function $\Psi$ is normalisable if

$$0<{\iiint }_{{\mathbb{R}}^3}|\Psi (\mathbf{x},t){|}^2{d}^{3}x<\infty ,\forall \text{ time }t$$

Normalised Wave Function

Wave function $\Psi$ is normalised if

$${\iiint }_{{\mathbb{R}}^3}|\Psi (\mathbf{x},t){|}^2{d}^{3}x=1,\forall \text{ time }t$$

One-Dimensional Version

$${\int }_{-\infty }^{\infty }|\Psi (x,t){|}^{2}dx=1$$

Distribution Function

Let $\rho (\mathbf{x})$ be the Probability Density Function then

Distribution Function is defined as

$$F(x)\equiv {\int }_{-\infty }^{\infty }\rho (y)dy$$

Correspondence Principle

Tendency of Quantum Results to approach Classical Theory for Large Quantum Numbers

Expectation Value of Function of Position

Let $f(\mathbf{x})$ be a function of position
Let $\Psi (\mathbf{x},t)$ be wave function satisfying Schrödinger Equation

Expectation Value of Function of Position $f(\mathbf{x})$ is

$${\mathbb{E}}_{\Psi }(f(\mathbf{x}))={\iiint }_{{\mathbb{R}}^3}f(\mathbf{x})|\Psi (\mathbf{x},t){|}^2{d}^{3}x$$

One Dimension Version

$${\mathbb{E}}_{\Psi }(f(x))\equiv {\iiint }_{\infty }^{\infty }f(x)|\Psi (x,t){|}^{2}dx$$

Probability of Measuring Energy of Particle

Suppose IVP for Schrodinger’s Equation for particle in box is normalised then

Probability of Measuring energy of particle to be $E_n$ is $|c_n{|}^2$

Proof

$$1={\int }_0^a|\Psi (x,t){|}^{2}dx=\sum _{m,n=1}^{\infty }\overline{c_m}{c}_n{e}^{-i(E_n-E_m)t/\hslash }{\int }_0^a\overline{{\psi }_m}(x){\psi }_n(x)dx=\sum _{n=1}^{\infty }|c_n{|}^2$$