Phy5645/Heisenberg Uncertainty Relation 3

Let us assume that a particle has the wavefunction,

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 (x)=\left (\frac{\pi }{a}\right )^{-1/4}e^{-ax^{2}/2}.}

We now wish to verify the Heisenberg Uncertanity Principle for this case. To do so, we need to find the uncertainties in position and 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 \Delta p=\sqrt{\left \langle {p^{2}} \right \rangle -\left \langle {p} \right \rangle ^{2}}} 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 \Delta x=\sqrt{\left \langle {x^{2}} \right \rangle -\left \langle {x} \right \rangle ^{2}}.}

Lets start by calculating the expectation values one by 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 \langle {x}\rangle =\left \langle {\Psi \left |{x} \right |\Psi } \right \rangle =\int\limits_{-\infty}^{\infty} {x\left |{\Psi (x)} \right |^{2}\,dx}=\sqrt {\frac{a}{\pi }} \int_{-\infty}^{\infty} xe^{-ax^{2}}\,dx=0 }

since the integrand is odd and thus the integral over all space is zero.

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 \left \langle {x^{2}} \right \rangle =\left \langle {\Psi \left |{x^{2}} \right |\Psi } \right \rangle } 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\limits_{-\infty }^{\infty } {x^{2}\left |{\Psi (x)} \right |^{2}\,dx}=\sqrt {\frac{a}{\pi }} \int_{-\infty}^{\infty} x^{2}e^{-ax^{2}}\,dx=\tfrac{1}{2}\sqrt {\frac{a}{\pi }} \sqrt {\frac{\pi }{a^{3}}} =\frac{1}{2a}}

Since the integral is of a Gaussian times a power 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 x} , we are able to use the known results for such integrals.

Similarly to 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 \left \langle {x} \right \rangle, } 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 \left \langle {p} \right \rangle =\left \langle {\Psi \left |{p} \right |\Psi } \right \rangle =0} because the integrand will be an odd function as well.

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 \left \langle {p^{2}} \right \rangle =\left \langle {\Psi \left |{p^{2}} \right |\Psi } \right \rangle }

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 =\sqrt {\frac{a}{\pi }} \int {e^{-ax^{2}/2}} \left (\frac{\hbar }{i}\frac{\partial}{\partial x}\right )^{2}e^{-ax^{2}/2}dx}

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 =\sqrt {\frac{a}{\pi }} (-\hbar ^{2})\int {e^{-ax^{2}/2}} \frac{\partial}{\partial x}\left (-\frac{2ax}{2}e^{-ax^{2}/2}\right )\,dx}

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 =\sqrt {\frac{a}{\pi }} (-\hbar ^{2})\int {e^{-ax^{2}/2}} \left \lbrace {-ae^{-ax^{2}/2}+\left ({-\frac{2ax}{2}} \right )\left ({-\frac{2ax}{2}} \right )e^{-ax^{2}/2}} \right \rbrace dx}

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 =\sqrt {\frac{a}{\pi }} (\hbar ^{2}a)\int {e^{-ax^{2}}} dx\text{ +}\sqrt {\frac{a}{\pi }} (-\hbar ^{2}a^{2})\int {x^{2}e^{-ax^{2}}} dx}

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 =\sqrt {\frac{a}{\pi }} (\hbar ^{2}a)\sqrt {\frac{\pi }{a}} +\sqrt {\frac{a}{\pi }} (-\hbar ^{2}a^{2})\frac{1}{2}\sqrt {\frac{\pi }{a^{3}}} }

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 \text{=}(\hbar ^{2}a)-\frac{\hbar ^{2}a}{2}=\frac{\hbar ^{2}a}{2}}

Combining these results, we obtain 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 \Delta p=\hbar\sqrt{\frac{a}{2}}} 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 \Delta x=\frac{1}{\sqrt{2a}}}

finally,

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 \Delta p\,\Delta x =\hbar\sqrt{\frac{a}{2}}\frac{1}{\sqrt{2a}} =\sqrt {\frac{\hbar ^{2}}{4}} =\frac{\hbar }{2}} .

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