Chapter 7 - Topic 7B - Dynamics of microscopic systems - Problems - Page 311: 7B.1

The normalized wave functions are $i) \space \big( {2\over L}\big)^{1\over 2}\times sin({n\pi x \over L})$ $ii) \space {1\over c \space (2L)^{1/2}}\times c$ $iii) \space \frac{1}{{(\pi a^3)}^{1\over 2}} \times e^{-r/a} $ $iv) \space\frac{1}{(32\pi a^5)^{1\over 2}} \times xe^{-r/2a}$

To normalize a wavefunction say $\psi_{(x)}$ in one dimension we need to find N such that $$N^2\int \psi^*_{(x)} \psi_{(x)} dx= 1$$ and then the normalized wavefunction is $N\psi_{(x)}$ $i) \space \psi = sin({n\pi x \over L}) \space for \space 0\le x\le L $ $N^2\int_0^L sin({n\pi x \over L}) sin({n\pi x \over L})dx = 1$ $N^2\int_0^L (sin({n\pi x \over L}))^2dx = 1$ ${N^2\over 2}\int_0^L \big( 1-cos({2n\pi x\over L}) \big)dx = 1$ ${N^2\over 2} \big[x-{sin({2n\pi x\over L}) \over {2n \pi \over L}} \big]_0^Ldx = 1$ ${N^2 \over 2}L = 1$ $N = \big({2\over L}\big)^{1\over 2}$ Hence Normalized wave function is $\big( {2\over L}\big)^{1\over 2}sin({n\pi x \over L})$ $ii)\space \psi_{(x)} =c \space for \space -L\le x \le L\quad (where \space \space c \space is \space constant)$ $N^2\int_{-L}^L c^2 dx = 1$ $N^2 c^2 [x]_{-L}^L = 1$ $2N^2 c^2 L = 1$ $N = {1\over c \space (2L)^{1/2}}$ Hence Normalized wave function is $ {1\over \space (2L)^{1/2}} $ $iii) \space \psi_{(r)} = e^{-r/a}$ $N^2\int_{vol.} e^{-2r/a} d \tau = 1$ $N^2\int_{vol.} e^{-2r/a} r^2 sin\theta dr d\theta d\phi = 1$ $N^2\int_{r=0}^{r=\infty} e^{-2r/a} r^2 dr \int_{\theta = 0}^{\theta = \pi}sin\theta d\theta \int_{\phi = 0}^{\phi = 2\pi}d\phi = 1$ $N^2 \times \frac{a^3}{4}\times 2 \times 2 \pi = 1$ $N = \frac{1}{{(\pi a^3)}^{1\over 2}}$ Hence Normalized wave function is $ \frac{1}{{(\pi a^3)}^{1\over 2}} e^{-r/a} $ $iv)\space \phi_{(r, \theta, \phi)} = xe^{-r/2a} = r sin\theta cos \phi e^{-r/2a}$ $N^2 \int_{vol.} r^2 sin^2\theta cos^2\phi e^{-r/a} d\tau = 1$ $N^2 \int_{vol.} r^2 sin^2\theta cos^2\phi e^{-r/a} r^2 sin\theta dr d\theta d\phi = 1$ $N^2 \int_0^{\infty} r^4 e^{-r/a}dr \int_0^{\theta}sin^3\theta d\theta \int_0^{2\pi} cos^2\phi d\phi = 1$ $N^2 \times 24 a^5 \times \frac{4}{3}\pi \times \pi = 1$ $N = \frac{1}{(32\pi a^5)^{1\over 2}}$ Hence the normalized wave function is $\frac{1}{(32\pi a^5)^{1\over 2}} xe^{-r/2a}$