Phy5645/Heisenberg Uncertainty Relation 3

Let's say that a particle has 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)=(\frac{\pi }{a})^{-1/4}e^{-ax^{2}/2}} and we are trying to verify Heisenberg Uncertanity relation.

In order to verify the uncertanity relation, we need to find these elements,

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 \vartriangle p=\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 \vartriangle x=\left \langle {x^{2}} \right \rangle -\left \langle {x} \right \rangle ^{2}} .

Lets start by calculating 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 \left \langle {x} \right \rangle =\left \langle {\Psi \left |{x} \right |\Psi } \right \rangle =\int\limits_{-\alpha }^{\infty } {x\left |{\Psi (x)} \right |^{2}dx}=\sqrt {\frac{a}{\pi }} \int {xe^{-ax^{2}}dx=0} } since it is an odd function and its integral over all the 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 {x^{2}e^{-ax^{2}}dx=} \sqrt {\frac{a}{\pi }} \frac{1}{2}\sqrt {\frac{\pi }{a^{3}}} =\frac{1}{2a}}

Since the integral is Gaussian integral, we used Gaussian integral results.

Just as 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 } , also 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 it is an odd function as well.

If we look at 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 } ,

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}} (\frac{\hbar }{i}\frac{\Game }{\Game x})^{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{\Game }{\Game x}(\frac{-2ax}{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 \text{=}\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 \text{=}\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 \text{=}\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}}

So, results are; 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 \vartriangle p=\left \langle {p^{2}} \right \rangle -\left \langle {p} \right \rangle ^{2}=\frac{\hbar ^{2}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 \vartriangle x=\left \langle {x^{2}} \right \rangle -\left \langle {x} \right \rangle ^{2}=\frac{1}{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 \sqrt {\vartriangle p\vartriangle x} =\sqrt {\frac{\hbar ^{2}a}{2}\frac{1}{2a}} =\sqrt {\frac{\hbar ^{2}}{4}} =\frac{\hbar }{2}} .

This is basic problem about an Uncertainty realtion. It basically provides that the more distribution we get around x, the smaller distribution we get around momentum and vice versa.

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