Chapter 39: Q39P (page 1216)

Verify that Eq. 39-44, the radial probability density for the ground state of the hydrogen atom, is normalized. That is, verify that the following is true: $\int_{0}^{\infty} P (r) dr = 1$

Short Answer

Expert verified

It is proved that $\int_{0}^{\infty} P (r) d r = 1$ .

Step by step solution

Radial probability density:

The expression of radial probability density is given by,

$P (r) = \frac{4}{a^{3}} r^{2} e^{- 2 \frac{r}{a}}$ ….. (1)

Here, $a = 52.292 \times 10^{- 12} m$ is the Bohr radius.

Prove that∫0∞P(r)dr=1:

Integrate the radial probability density from zero to infinity.

$\int_{0}^{\infty} P (r) d r = \int_{0}^{\infty} \frac{4}{a^{3}} r^{2} e^{- 2 \frac{r}{a}} d r = \frac{4}{a^{3}} \int_{0}^{\infty} (r^{2} e^{- 2 \frac{r}{a}}) d r = \frac{4}{a^{3}} {(- r^{2} (\frac{a}{2}) e^{- 2 \frac{r}{a}} - \frac{a^{2} r}{2} e^{- 2 \frac{r}{a}} - \frac{a^{3}}{4} e^{- 2 \frac{r}{a}})}_{0}^{\infty} = \frac{4}{a^{3}} {(- \frac{r^{2}}{2} e^{- 2 \frac{r}{a}} - \frac{a r}{2} e^{- 2 \frac{r}{a}} - \frac{a^{2}}{4} e^{- 2 \frac{r}{a}})}_{0}^{\infty}$

Simplify further:

$\int_{0}^{\infty} P (r) d r = \frac{4}{a^{2}} (0 - 0 - {|\frac{a^{2}}{4} e^{- \frac{2 r}{a}}|}_{0}^{\infty}) = - \frac{4}{a^{2}} (\frac{a^{2}}{4} e^{- \frac{2 \infty}{a}} - e^{0}) = 1$

Hence, it is proved that . $\int_{0}^{\infty} P (r) d r = 1$

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Verify that Eq. 39-44, the radial probability density for the ground state of the hydrogen atom, is normalized. That is, verify that the following is true: $\int_{0}^{\infty} P (r) dr = 1$

Short Answer

Step by step solution

Radial probability density:

Prove that∫0∞P(r)dr=1:

One App. One Place for Learning.

Most popular questions from this chapter

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Study anywhere. Anytime. Across all devices.

Verify that Eq. 39-44, the radial probability density for the ground state of the hydrogen atom, is normalized. That is, verify that the following is true:∫0∞P(r)dr=1

Short Answer

Step by step solution

Radial probability density:

Prove that∫0∞P(r)dr=1:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Oscillations

Quantum Physics

Atoms and Radioactivity

Electricity and Magnetism

Electromagnetism

Torque and Rotational Motion

Study anywhere. Anytime. Across all devices.

Verify that Eq. 39-44, the radial probability density for the ground state of the hydrogen atom, is normalized. That is, verify that the following is true: $\int_{0}^{\infty} P (r) dr = 1$