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A probability amplitude is a complex quantity used to describe system behaviour in quantum mechanics.

Use the formula,$P={鈭�}_{x_1}^{x_2}{y_0}^{*}{y}_{0}dx$

Here, $y_0$represent the wave function of the particle and $P$is the probability

Born's interpretation is that ${\left|{唯}_0(x,t)\right|}^2$represents the probability distribution for the particle's position in the ground state at time t. The likelihood that a particle will be found in a location that is traditionally permissible is ${鈭�}_{-a}^a鈭�{唯}_0(x,t{)}^{2}dx$.

The probability that it鈥檚 not in this region is,

$1-{鈭�}_{-a}^a{\left|{唯}_0(x,t)\right|}^{2}dx=1-{鈭�}_{-a}^a{唯}_0(x,t){唯}_0^{*}(x,t)dx$

localid="1660906664910" $=1-{鈭�}_{-a}^a\left[-{蠄}_0\left(x\right)e^{-E_{0}t/\mathrm{鈩�}}\right]\left[{蠄}_0\left(x\right)e^{i{E}_{0}t/\mathrm{鈩�}}\right]dx$

$=1-{鈭�}_{-a}^a{\left[{蠄}_0\left(x\right)\right]}^{2}dx$

localid="1660906677303" $=1-{鈭�}_{-a}^a{\left[{\left(\frac{m蝇}{蟺{\displaystyle \mathrm{鈩�}}}\right)}^{1/4}\exp \left(-\frac{m蝇}{2{\displaystyle \mathrm{鈩�}}}{x}^2\right)\right]}^{2}dx$

Further evaluating,

localid="1660906688274" $=1-{鈭�}_{-a}^a\left[{\left(\frac{m蝇}{蟺{\displaystyle \mathrm{鈩�}}}\right)}^{1/2}\exp \left(-\frac{m蝇}{{\displaystyle \mathrm{鈩�}}}{x}^2\right)\right]dx$

localid="1660906698298" $=1-\sqrt{\frac{m蝇}{{\displaystyle \mathrm{鈩�}}}{鈭�}_{-a}^a}\exp \left(-\frac{m蝇}{{\displaystyle \mathrm{鈩�}}}{x}^2\right)dx$

Use

localid="1660906707184" $尉=\sqrt{\frac{m蝇}{{\displaystyle \mathrm{鈩�}}}}x$

localid="1660906713573" $d尉=\sqrt{\frac{m蝇}{{\displaystyle \mathrm{鈩�}}}}dx$

localid="1660906720474" $dx=\sqrt{\frac{{\displaystyle \mathrm{鈩�}}}{m蝇}}d尉$

Thus,

localid="1660906740360" $1-{鈭�}_{-a}^a{\left|{蠄}_0\left(x,t\right)\right|}^{2}dx=1-\sqrt{\frac{m蝇}{蟺{\displaystyle \mathrm{鈩�}}}}{鈭�}_{-\sqrt{\frac{m蝇}{{\displaystyle \mathrm{鈩�}}}}}^{\sqrt{\frac{m蝇}{{\displaystyle \mathrm{鈩�}}}}}{e}^{-{尉}^2}\left(\sqrt{\frac{{\displaystyle \mathrm{鈩�}}}{m蝇}}d尉\right)$

localid="1660906752800" $=1-\frac{1}{\sqrt{蟺}}{鈭�}_{-\sqrt{\frac{m蝇}{{\displaystyle \mathrm{鈩�}}}}}^{\sqrt{\frac{m蝇}{{\displaystyle \mathrm{鈩�}}}}}{e}^{-{尉}^2}d尉$

localid="1660906763118" $=1-\frac{2}{\sqrt{蟺}}{鈭�}_0^{\sqrt{\frac{m蝇}{{\displaystyle \mathrm{鈩�}}}}}{e}^{-{尉}^2}d尉$

Now the error function is defined as

localid="1660906772839" $1-{鈭�}_{-a}^a{\left|{唯}_0(x,t)\right|}^{2}dx=1-erf\left(\sqrt{\frac{m蝇}{{\displaystyle \mathrm{鈩�}}}}a\right)$

Thus the probability is,

localid="1660906782708" $1-{鈭�}_{-a}^a{\left|{唯}_0(x,t)\right|}^{2}dx=1-erf\left(\sqrt{\frac{m蝇}{{\displaystyle \mathrm{鈩�}}}}a\right)$

localid="1660906791838" ${蠄}_0={\left(\frac{m蝇}{蟺{\displaystyle \mathrm{鈩�}}}\right)}^{\frac{1}{4}}{e}^{-{尉}^2/2}$

The energy of the ground state is known, so

localid="1660906804773" $\frac{{\displaystyle \mathrm{鈩�}}蝇}{2}=\frac{1}{2}m{蝇}^2{a}^2$

Evaluate for a and get,

localid="1660906820214" $a=\sqrt{\frac{{\displaystyle \mathrm{鈩�}}}{m蝇}}$

Thus the probability is,

$1-{\left.{鈭�}_{-i}^a{唯}_0(x,t)\right|}^{2}dx=1-erf1$

$鈮�0.157$

Therefore, the probability that the particle lies outside the classically allowed region in the ground state is $0.157$.